Here’s your next reading assignment. Read Sections 4.3.1-4.3.3 in your textbook and answer the following questions by 8 a.m., Friday, March 2nd. Be sure to login (using the link near the bottom of the sidebar) to the blog before leaving your answers in the comment section below.
- Water samples are taken from water used for cooling as it is being discharged
from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most 150 degrees Fahrenheit, there will be no negative effects on the river’s ecosystem. To investigate whether the plan is in compliance with the regulations that prohibit a mean discharge water temperature above 150 degrees, 50 water samples will be taken at randomly selected times, and the temperatures of each sample recorded. Identify an appropriate null and alternative hypothesis for this test, and justify your answer. - Suppose, in the context of the previous context, the average temperature for the 50 water samples collected was 152 degrees. Why is it too simple just to say that since 152 is greater than 150, the power power plant discharge temperature is higher than it should be?
- What’s one question you have about the reading?
1) H0: The mean discharge water temperature is 150 degrees Fahrenheit.
HA: The mean discharge water temperature is above 150 degrees Fahrenheit.
Since the river’s ecosystem will be adversely effected only if temperature of the discharge water rises above 150 degrees F, the null should be that the mean is 150 degrees F (or less), while the alternate hypothesis should be that the mean is greater than 150.
2) There might have been variation in the sample which would account for the sampling mean to be higher than 150 degrees F. This is not grounds to state the power plant discharge temp is higher than it should be, since the observations in the sample might have just happened to be the observations with higher temps than other observations.
3) Can the null hypothesis be of the form, “the mean is less than or equal to” as opposed to just “the mean is equal to”?
1. The null hypothesis is that the mean temperature of the water discharged by the river is less than or equal to 150 degrees Fahrenheit. This is the null hypothesis because this is the claim we are interested in testing. The alternative hypothesis is that the mean temperature is above 150 degrees. This alternative hypothesis represents a new claim, usually that there has been some kind of change in the data.
2. Just because the average of that sample is 152 degrees does not mean that the actual mean temperature of the water is more than 150 degrees. Samples approximate the true values of the population. For this sample, a confidence interval constructed would probably include 150 and some values less than 150, meaning that it is reasonable for the true value to be less than or equal to 150. We could have somehow managed to take the 50 hottest water samples collected over the course of the test, so their average is a lot higher than the actual mean.
3. On page 157, the text says “Even if we fail to reject the null hypothesis, we typically do not accept the null hypothesis as true.” In a problem like the water sample problem above, where either the null hypothesis or the alternative hypothesis logically has to be true, how is it that we can not accept the null hypothesis as true?
1.) Ho = Mean of temperature of 50 water samples is less than 150 degrees.
Ha = Mean of temperature of 50 water samples is not less than 150 degrees.
2.) This is too simple because there may be one or two major outliers pulling up the mean while the others fall in the right interval.
3.) What happens when Ha is only one other possibility, cant it be referred to as Ho also?
1.
An appropriate null hypothesis is that the mean plant water discharge temperature is below 150 degrees. This is the null hypothesis because it represents the skeptical side of the experiment, doubting that the water is actually dangerous to the ecosystem. An appropriate alternative hypothesis is that the mean plant water discharge temperature is above 150 degrees. This can be considered alternative because it represents a claim different from the skeptics point of view. Also, for a skeptic to reject that the water is below 150 degrees, it must be strongly over 150 degrees.
2.
The 152 degrees measurement only comes with a certain amount of confidence associated with it, therefore one cannot definitively say that the mean power plant water temperature is higher than it should be. One can only say that the mean water temperature is likely to be 152 degrees with a certain amount of confidence.
3.
In example 4.22, we cannot reject the null hypothesis because is falls within the confidence interval, but does the fact that it is extremely unlikely that the means are the exact same ever come into account when we make a statement about the likelihood of the two events being the same or different?
1. Ho=The mean temperature of discharged water is not above 150 degrees F. (“innocent”)
Ha= The mean temperature of discharged water is above 150 degrees F. (“guilty”)
If the water temperature is significantly different than 150 degrees on the low side, it has no effect on compliance. The test has to be one sided because the only way the plant would be “guilty” is is the temperature is significantly greater than 150 degrees.
2. Every time you take a sample the average will be a little bit different. Without knowing the standard deviation of the data there is no way to know how significant a two degree difference really is. If the sample was taken again, the temperature could be higher or lower and this variance has to be accounted for using a confidence interval.
3. I didn’t really understand the purpose of the part about type 1 and type 2 errors. People make them and if try to stop one kind you make the other kind. How is that helpful?
1) The null hypothesis is that the mean water temperature is below 150 degrees and the alternate hypothesis is that the mean is above 150 degrees. That is because the alternate hypothesis is what you want to test, so in this experiment we are testing to see if the mean water temperature is above 150 degrees; therefore that’s the alternate hypothesis.
2) It’s too simple to rule out the null hypothesis based off of that mean because if you use confidence intervals, 152 degrees could still be in the range of the confidence intervals. That means that you can’t be completely sure that the null hypothesis is incorrect but you also can’t be completely sure that it is correct.
3) What further testing is used if the data isn’t strong enough to prove one hypothesis correct?
1. A null hypothesis is the the temperature of the water is less than or equal to 150 degrees and is within accepted standards. An alternative hypothesis is that the temperature is above the accepted standard of 150 degrees. The null hypothesis is valid because, while it cannot be proven, it can be accepted if the data supports it. The same logic supports the alternative hypothesis.
2. Without looking at raw data or the values of standard deviation and etc., the difference from the accepted standard is far too small to conclude that the plant is discharging overly hot water.
3. How would you test for errors in a sample like the running speed example or like in the scenario above?
1. This depends on if you want to err on the side of caution or not. My null Hypothesis would be that the mean is over 150 degrees Fahrenheit, and the alternative Hypothesis would be that the mean is under 150 degrees. This is because I want to be “skeptical” of the river not being affected, since that is the least dangerous outcome.
2. Because we are taking sample means, the average of those means does not necessarily represent the population mean. This is why we us confidence intervals.
3. The reading made sense.
1) The null hypothesis would be that the mean of the 50 samples would be above 150 degrees fahrenheit, and the alternative hypothesis would be that the mean of the 50 samples is 150 degrees fahrenheit or less.
2) It is too simple to say that the power plant is violating the regulation because a 2 degree difference from the expected value is within a 95% confidence interval, so the difference is not large enough to accept the null hypothesis.
1) H0 = The water samples will be greater than 150 degrees
HA = The water samples will comply with the regulation and be under 150 degrees
2) We don’t know what the confidence interval is. For all we know, the samples collected were higher than usual. We can’t completely confirm the hypothesis if 150 degrees is contained in the confidence interval for the samples.
3) I’m having a bit of trouble constructing a good null hypothesis. The text didn’t really explain criteria for doing this, and in setting up an experiment choosing the right null hypothesis is everything
1. Null Hypothesis: The water temperature is above the accepted temperature required to be in compliance with the regulations.
Alternative Hypothesis: The water temperature is within the regulated range.
The test is done to make sure that the water temperature is in compliance. Thus, these hypothesis should be true.
2. You have to look at the standard deviation and mean and conduct some kind of statistical test (z-test) in order to prove that the statistic is valid to make an assumption.
3. The reading was straight forward.
1. Null hypothesis: The mean temperature of the discharged water is above 150 degrees.
Alternative hypothesis: The mean temperature of the discharged water is not above 150 degrees.
The two hypothesis cannot be ture at the same time.
2. Since 152 is very close to 150 and the sample size is fiarly small, we can only say that we fail to reject the null hypothesis.
3. To what extent should we use double negative conclusion instead of acceptinging a hypothesis.
1. null hypothesis: mean temperature of the discharged water is higher than 150 degrees Fahrenheit.
alternative hypothesis:mean temperature of the discharged water is lower than 150 degrees Fahrenheit.
2.The observed mean value may be way off the true distribution.
1) Null hypothesis: The mean of 50 samples = 150 Fahrenheit.
Alternative hypothesis : The mean of 50 samples is not 150 Fahrenheit.
The null hypothesis is that the plant is operating according to regulation, therefore the mean of the sample should be at most 150 degrees.
2) Since the samples are collected at randomly selected time, the surrounding temperature might have an effect on the samples. The temperature of a sample collected at night would be less compared to one collected in the afternoon. Therefore, 152 degrees might be in the confidence interval that contain 150 degrees.
3) Does lower significance level makes it harder to reject null hypothesis?
1) The null hypothesis is that the average number of water samples will be below 150 degrees. The alternative hypothesis is that the water samples will have an average above 150 degrees with enough confidence to say that the water is above 150 degrees. We want to be sure that the number is above 150 degrees before we shut down the plant.
2) There is some variance in the mean taken from multiple samples. A confidence interval would need to be determined to decide if we are willing to decide that this is significant evidence. The limiting factor here is the variance, which may be too high to gauge if this is significant evidence.
3) Does it make sense to have multiple alternative hypothesis options on the same parameter.
1) Null hypothesis would be that the water is discharging at the correct temperature, because we are assuming that the system works properly and are looking for errors. The alternative hypothesis is that it is discharging at the incorrect temperature, because this represents a deviation from the expected behavior.
2) We don’t know about the variation of the samples, so the correct temperature could be in the 95% confidence interval of the sample mean temperature and thus we can’t make a determination about the validity of the null hypothesis.
3) None, it was very straightforward!
The null hypothesis: average temp of water samples will be 150 degrees
If there is enough confidence in the alternative hypothesis, water may be assumed to be warmer than 150 degrees.
The water should be below 150. So the null hypothesis is that it will be below 150.
Here we are taking data from a sample and trying to make assumptions about all the water. When we take the mean of the samples we have to remember that there will be some variance and that a confidence level must be established. An average of 50 samples so close to 150 degrees is not enough to make the assumption.
Maybe we can practice assigning the null and alternative hypothesis?
1. Water samples are taken from water used for cooling as it is being discharged from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most 150 degrees Fahrenheit, there will be no negative effects on the river’s ecosystem. To investigate whether the plan is in compliance with the regulations that prohibit a mean discharge water temperature above 150 degrees, 50 water samples will be taken at randomly selected times, and the temperatures of each sample recorded. Identify an appropriate null and alternative hypothesis for this test, and justify your answer.
H0 = water sample is at or below 150 degrees
HA = water sample is above 150 degrees
2. Suppose, in the context of the previous context, the average temperature for the 50 water samples collected was 152 degrees. Why is it too simple just to say that since 152 is greater than 150, the power plant discharge temperature is higher than it should be?
150 is within the range of plausible values, so we cannot say the null hypothesis is impossible.
3. What’s one question you have about the reading?
How would you know if you committed an error?
1) Null hypothesis, Ho = mean temperature of sampled discharge water is less than or equal to 150 degrees
Alternative hypothesis, H_A = mean temperature of sampled discharge water is more than 150 degrees
Null hypothesis represents a perspective of no difference, which means that the mean temperature of sampled discharge water is not more than is allowed. Prohibition is in effect when the temperature gets above 150 degrees, and this change is represented by the alternative hypothesis.
2) The variations between samples may make our estimate get close but not exactly equal to the parameter. If we take a 100 water samples, the average would not exactly be the same, but we might get closer to its actual value with larger sample set. To reject the null hypothesis, and accept the alternative that the average temperature is more than 150 degrees ( = 152 degrees), we should set up a confidence interval, and see if the null parameter falls into the possible range or not. If it is, then we cannot reject Ho, but we can disregard H_A. If it is not, then we can reject Ho and accept H_A as true.
3) Can you explain more on the significance level, alpha? Do we use the population standard deviation, delta, or the sample standard deviation, s in our calculation?
1. Null Hypothesis : The temperature is at most 150ºF (claim to be tested)
Alt Hypothesis: The temperature is above 150ºF (alternate claim)
2. We have to create a confidence interval and see if that value lies in the interval.
3. Pretty short and straight forward!
1) We can take the null hypothesis when the average number of samples is smaller than 150 degrees and the alternative when it’s higher than 150. If it’s higher than 150 we shut down the plant.
2) We can make a judgment by looking at the confidence interval and deciding whether or not the difference is significant. It may be that it isn’t
3) What if you come up with more than one alternative hypothesis?
1) Ho: the mean temperature of the discharged water is at most 150 degrees Fahrenheit.
Ha: the mean of temperature of the discharged water exceeds 15o degrees Fahrenheit.
If we would get enough evidence to reject Ho in favor of Ha, then we would have enough evidence to conclude that there will be no negative effects on the river’s ecosystem.
2) Variability always exists and deviations are possible. We don’t just compare sample and population means and make our decisions based on that. What we are looking for is finding a probability of getting a result as extreme as a given one.
3) I see that lowering Type I Error would increase Type II Error and lowering Type II Error would increase Type I Error. What would be a good compromise? And are there situations in which we actually want as low Type I Error or Type II Error as possible without caring that it increases the other one?
1. Null: The average temperature for the 50 water samples is below 150 degrees. This represents the claim that is being tested, in this case, the temperature of the water. Alternative: The average temperature for the 50 water samples will vary between 120 and 150 degrees. This represents a claim with a range of values.
2. It is not accurate to reject hypothesis unless there is clear evidence to. We need to reduce error so that the hypothesis does not fail.
3. How do you come up with a good null hypothesis?
1– The null hypothesis in this case is :the mean # of water samples below 150 degrees. An alternative hypothesis would be the water samples will have an average o fmore than 150 degrees, but with a high enough confidence that we can say that the water is above 150 degrees. We have to be certain before the plant is shut down.
2) There aren’t enough samples taken to say that it would be a valid sample size.
3) none
1. H0 = The mean temperature of the discharged water 150 degrees Fahrenheit, and it does not meet regulations. This is become we naturally assume that the mean temperature meets regulations and are skeptical that it does not, meaning that the alternative hypothesis is when it does not.
2. This is too simple because we have to take into account the fact that there is a degree of uncertainty with the measurements. They are not precise, and they do not give us perfect values.
3. I have no questions about the reading; I have seen this material before.
1)null-the average of the sample mean of the 50 water taken. Alternative, the difference between the highest number vs the lowest number of all 50 data taken.
2) No we can’t easily say that because the sample size is not big enough! plus, to make sure it is certain we have to do the confidence interval test and make sure that all of that fall in the 95% interval value and if not we can’t just SIMPLY say it is more than 150.
3)What is the difference between null and alternative?
1. Null hypothesis: mean discharge temperature is not above 150 degrees
Alternative hypothesis: mean discharge temperature is above 150 degrees
Since the null hypothesis is a kind of ‘default’ hypothesis, a safe/standard discharge temperature seemed a better fit.
2. We only have 50 water samples, which could be biased.
3. Today’s reading reminded me of a question I had about the last reading we did. How can you calculate an appropriate z-score to get any confidence level you want? We touched on this in class.. it may be in my notes…
1. H0 150
use confidence intervals to predict whether the temperature is above 150
2. there may be a small uncertainty that the sample mean may be from an interval outside the true mean.
I don’t understand null and alternative hypothesis
it’s too simple to say that the water temperature is higher than it should be because there are only 50 samples
Water samples are taken from water used for cooling as it is being discharged
from a power plant into a river. It has been determined that as long as the mean temperature of the discharged water is at most 150 degrees Fahrenheit, there will be no negative effects on the river’s ecosystem. To investigate whether the plan is in compliance with the regulations that prohibit a mean discharge water temperature above 150 degrees, 50 water samples will be taken at randomly selected times, and the temperatures of each sample recorded. Identify an appropriate null and alternative hypothesis for this test, and justify your answer.
H0: Mean discharge temperature 150 degrees
The answer comes from the statement “To investigate whether the plan is in compliance with the regulations that prohibit a mean discharge water temperature above 150 degrees,” which suggests that “compliance” refers to the null hypothesis of 150.
Suppose, in the context of the previous context, the average temperature for the 50 water samples collected was 152 degrees. Why is it too simple just to say that since 152 is greater than 150, the power power plant discharge temperature is higher than it should be?
We do not know the variance in the temperature distribution. Suppose the variation is quite large (say 20 degrees is one std dev). Then a temp of 152 is just a single sample in some larger distribution. You must consider how far is that sample from the mean in terms of the std deviation and what distribution is the sample probably from. If the std dev of the set is very large, then a temp of 152 can be from a number of distribution (where each distribution is the same shape, but simply shifted along the mean since thats what we are testing for). So if H0 says we have a distribution centered around 150, and HA says we have a distribution centered around a value above 150, you can’t possible know if a single sample 152 resides in the former distribution or the latter (especially if the distribution is wide, i.e. large variation in the data).
What’s one question you have about the reading?
Practice problems needed
Review how the confidence interval is being used for H0 and HA
The null hypothesis is that the mean temperature of the water samples is at most 150 degrees, because the null hypothesis is “the claim to be tested.” The alternative hypothesis is that the mean exceeds 150 degrees because it represents the “alternative claim under consideration.”
Suppose, in the context of the previous context, the average temperature for the 50 water samples collected was 152 degrees. Why is it too simple just to say that since 152 is greater than 150, the power power plant discharge temperature is higher than it should be?
The samples that were collected might be misleading in that they don’t reflect the real mean. If it was a sunny day, for example, the surface temperature of the water could have raised. If this is where the samples were drawn from, it could have greatly effected the mean.
I know they said the goal of this type of display was to cater to the skeptics. Who else is it aimed for?
The null hypothesis for this scenario is that the plan is in compliance with the regulations, and the alternative hypothesis is that the plan is not in compliance. This is justified by the definitions of the null and alternative hypotheses. A null hypothesis is a skeptical claim to be tested. Since we are checking compliance, that is the null hypothesis, that we are complying with regulations. The alternative hypothesis is the other claim under consideration (failing), and can be represented with the temperature samples.
The regulation mean (150 degrees) could easily fall in the range of plausible values for the measured mean, depending on the confidence interval for the experiment. We cannot simply say the null hypothesis (compliance) is implausible because the calculated mean is greater than 150. We must be for sure out of the range of 150 degrees (and therefore in the alternative hypothesis entirely) to reject the null hypothesis.
Are there more hypotheses depending on the number of variables that can be in effect for a given system?
1. The null hypothesis is that if the mean of the temperatures of the plant’s discharges is 150˚F or less, it will not disrupt the ecosystem. The alternative hypothesis is that the mean temperature of the discharges can be a temperature greater than 150˚F and it will not disrupt the ecosystem.
2. Since the 150˚F mean is the null hypothesis, we can be skeptical about it. Therefore, it is possible that a mean temperature of 152˚F could not be disruptive of the ecosystem.
3. Why is the null hypothesis named so? It seems like it’s more of a base hypothesis.
1. H0: x-bar 150 (what we fear it might be)
2. There is some variation associated with the samples take (ie happened to choose the 50 hottest samples). A confidence interval is more appropriate. Personally it seems the maximum value of the confidence interval should be 150 because anything higher hurts the environment.
3. No questions
1) Null is the water is at the correct temperature to not damage the ecosystem. Alternative is that the water is above the allowable 150 degrees and is damaging the ecosystem. Samples are random so it boils down to those who believe there isn’t a difference and those who believe there is.
2) 2 degrees off is not high enough for alarm. This could come down to aspects such as measurement error or inaccuracy, a surprisingly long week, etc. The evidence for the argument that the temperature is too high is not nearly strong enough.
3) No questions this time
1. The null hypothesis is that the average temp of water samples will be below 150 degrees. The alternative hypothesis is that the average temp of water samples will be above 150 degrees.
2. We don’t know the variance of this mean. Therefore we can not know the confidence interval clearly.
3. Why we have only one alternative hypothesis instead of multiple alternative hypothesis?
1. null hypothesis- the mean water temperature is higher than 150 degrees. Alternative- The mean water temperature is 150 or lower. The null hypothesis represents a skeptical perspective, while the alternative hypothesis represents and alternative claim under consideration.
2. Because this can be expressed as a double negative. And double negatives can be used to communicate that while we are not rejecting a position (that the power plant water temperature is too high), we are also not saying it is correct.
3. What does p-value express in layman’s terms? As the book definition wasn’t very helpful.
1. A null hypothesis could be that that the current season (winter vs summer) could increase powerplant production and therefore increase overall temperature. With basic knowledge of how powerplants work, this arguement is easy to see that an increased demand for power could surge the power over the average. The alternative hypothesis would be that both summer (large AC consumption) and winter (large heater) have the same amount of electrical demand and therefore a negligible difference.
2. We didn’t use a confidence interval! 152 could have been a fluke, as we would expect some sort of variation in the data. Having a confidence interval could yield the statement “we are 95% confident that the powerplant is acting illegally with this data” if it were under 152.
3. Nothing really
1. An appropriate null hypothesis would be that the average temperature of water samples will fall equal to or below 150 degrees F. An appropriate alternative hypothesis would be that the average temperature of water samples is greater than 150 degrees F. We want to make sure that there are no negative effects on the river’s ecosystem, which means that the average water samples are at or below 150 degrees F, in this example. The null hypothesis is essentially describing the situation in which there is no negative effect on the river’s ecosystem while the alternative hypothesis describes the situation in which there is a negative effect.
2. We need to calculate a confidence interval which requires a known standard deviation of either the entire population or of the sample population. Also, without the standard deviation, it is impossible to tell if a difference of two degrees is within the known deviations of the samples. It could very well be possible that the standard deviation is more than just a few degrees, in which case 152 degrees F would be perfectly acceptable. Without the standard deviation or a confidence interval, however, we cannot be sure.
3. Does an experiment have to be a controlled experiment in order to perform hypothesis testing? Based on what I recall from previous math courses, I do not think an experiment has to be controlled, but I have no clue why that is the case.
1. The null hypothesis would be that the mean temperature of the discharged water is at most 150 degrees farenheight. The alternative hypothesis is that the mean temperature is above 150 degrees farenheight.
2. A reading of 152 degrees farenheight could be due to sampling variation. In general, it is unlikely that the sample mean will be equal to the parameter itself. More samples would be needed in order to make this claim with better certainty.
3. Which standard deviation is used to calculate a confidence interval used for hypothesis testing?
1.) Null Hypothesis: The average temperature of the 50 samples will be less than 150 degrees.
Alternative hypothesis: The average temparature of the 50 samples will be greater than/equal to 150 degrees.
2.) There could be big outliers in the data driving up the mean temperature. As such, one cannot assume the temperature is higher.
3.) No questions.
1) Null hypothesis: average of water samples 150 degrees.
We want to check and be sure the water is 150 degrees.
3) Why do we bother to define significance levels when we already have a confidence level which is just a percentage equivalent of 1-? We don’t gain any more information by doing this do we?
I don’t understand what happened there, but I’m resubmitting this response because for some reason when i posted the comment, it removed my answer to the second question?
1) Null hypothesis: average of water samples 150 degrees.
We want to check and be sure the water is 150 degrees.
2) This doesn’t take variance within samples into account.
3) Why do we bother to define significance levels when we already have a confidence level which is just a percentage equivalent of 1-? We don’t gain any more information by doing this do we?
Oh come ON! I also just realized that it removed my answer what the alternative hypothesis was for question 1 and also my >, <= comparison signs.
Null hypothesis: average of water samples 150 degrees.
I give up. It keeps deleting lines in my responses. I’m just emailing my answer to Dr. Bruff.
A)
It most likely that the average is going to below 150 F. However, If we consider the hypothesis way, we would say the average will be above 150 F with enough confidence.
B)
Because the average was taken randomly from different samples. Therefore, the chance of the average is above 150F is high. Therefore, we can easily say that it is above the temperature is above 150 F in the planet.
C)
Can you accept the answer? (even I am late)