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The Clearwater Power Company produces electrical power from coal. A local environmental group claims that Clearwater's emissions have raised sulfur dioxide levels above permissible standards in Blue Sky, the town downwind of the plant.

According to Environmental Protection Agency standards, an acceptable average sulfur dioxide level is 30 parts per billion (ppb). As Clearwater's PR consultant, you want to defend the company, and you try to anticipate the environmentalist's argument.

The environmental group collects 36 samples on randomly selected days over the course of a year. It finds a mean sulfur dioxide content of 35 ppb with a standard deviation of 24 ppb.

The environmentalist group will use a hypothesis test to back up its claim that the sulfur dioxide levels are higher than permitted. Which of the following is an appropriate null hypothesis for this problem?

a. The average sulfur dioxide level is no higher than 30 ppb, the EPA's standard of acceptability. This is the best answer. The null hypothesis states the conventional wisdom: that the population mean of the population under investigation the sulfur dioxide concentration of air in Blue Sky is less than or equal to 30 ppb, the acceptability standard for which the EPA does not require a remedy. The environmentalists will pose as the alternative hypothesis the claim they are trying to substantiate: that Blue Sky's levels exceed the acceptable standard.

b. The average sulfur dioxide level is higher than 30 ppb, the EPA's standard of acceptability. This is not the best answer. This would make a good alternative hypothesis, since this is the claim the environmentalists are trying to substantiate.

c. The average sulfur dioxide level is 35 ppb.

This is not the best answer. The null hypothesis should state that the population mean takes on the intial value, not the value of the sample mean.

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The environmentalists' claim is that sulfur dioxide levels are higher, so they will want to run a one- sided test. The alternative hypothesis states that the sulfur dioxide levels are above the accepted standard. We assume they will choose a 95% confidence level.

What is the range of likely sample means? a. All values above 22.16 ppb.

This is not the correct answer. In a one-sided test, the z-value for 95% confidence is 1.645, the cumulative probability for 0.95.

b. All values above 23.42 ppb.

This is not the correct answer. We want to test for positive change, not negative change. c. All values below 36.58 ppb.

This is the correct answer. d. All values below 37.84 ppb.

This is not the correct answer. In a one-sided test, the z-value for 95% confidence is 1.645, the cumulative probability for 0.95.

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Utility for Single Populations

They calculate the one-sided range around the null hypothesis mean that contains 95% of all samples. The z-value for a one-sided 95% range is 1.645. The upper bound on the range of likely sample means is 36.58 ppb.

Based on your calculations, you should:

a. Reject the null hypothesis. The data indicate that the sulfur dioxide content in Blue Sky is above EPA standards.

This is not the correct answer. You reject the null hypothesis only when the sample mean falls outside the range of likely sample means.

b. Accept the alternative hypothesis.

This is not the correct answer. Accepting the alternative hypothesis is equivalent to rejecting the null hypothesis. You can reject the null hypothesis only when the sample mean falls outside of the range of likely sample means. c. Accept the null hypothesis.

This is not the correct answer. Since the sample mean falls within the range of likely sample means, you cannot reject the null hypothesis. However, this does not mean that you have proven that the null hypothesis is true. We never accept a null hypothesis based on sample data.

d. Do not reject the null hypothesis.

This is the correct answer. 35 ppb falls within the range of likely sample means. Ata a 95% confidence level, these sample data do not provide enough evidence to reject the null hypothesis.

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Utility for Single Populations Exercise 3: Neshey's Smooches

You are the plant manager of a Neshey's chocolate factory. The shop was flooded during the recent storms. The machine that wraps Neshey's popular chocolate confection, Smooches, still works, but you are afraid it may not be working at its former capacity.

If the machine isn't working at top capacity, you will need to have it replaced. Which type of hypothesis test is most appropriate for this problem?

a. One-sided test

This is the best answer. You want to know if the machine's performance has been impaired, not simply if the performance has changed.

b. Two-sided test

This is not the best answer. You use a two-sided test when testing for change in either direction.

The hourly output of the machine is normally distributed. Before the flood, the machine wrapped an average of 340 Smooches per hour. Over the first week after the flood, you counted wrapped

Smooches during 32 randomly selected one-hour periods. The machine averaged 318 Smooches per hour, with a standard deviation of 44.

You conduct a one-sided hypothesis test using a 95% confidence level. According to your calculations, you should:

a. Have the machine replaced.

This is the correct answer. The sample mean falls below the lower bound of the one-sided range of likely sample means around the null hypothesis mean. You can be 95% confident that the machine's performance has been impaired.

b. Continue to use the machine. The lower output in the sample hours you observed was due solely to chance.

This is not the correct answer. Be sure you correctly calculated the lower bound of the range of likely sample means to be 327.

The null hypothesis is that # " 340. The alternative hypothesis is that # < 340 since you are using a one-tail test and you are assuming that the new population mean is lower than the population mean before the flood.

Identify the relevant values. The sample size n=32. The standard deviation s=44. The appropriate z- value is 1.645 if you want to capture 95% of all sample means in a one-sided range around the null hypothesis mean.

Use the formula and calculate the lower bound, 327. The sample mean of 318 falls well outside of the calculated range of likely sample means. You accept this as strong evidence against the null

hypothesis, substantiating the alternative hypothesis that the mean output rate has dropped. You should replace the machine.

Single Population Proportions

Happy with your work on restaurant spending, Leo jumps right into the next problem. "It's not just the revenue of the restaurants that I care about," Leo says, "It's also my guests' satisfaction with their restaurant experience."

The Restaurant Ambiance Problem

When I go out to eat, I expect more than just excellent food. The whole dining experience is essential — everything from the service, to the décor, to the design and quality of the silverware.

And it's not just that all of these factors must be excellent individually — they have to fit together. The restaurant has to have ambiance! I'm sure my guests have similar expectations, and I want to be sure my restaurant meets them.

Since my new chef introduced more sophisticated cuisine, I made some changes to the décor that I think have improved the ambiance.

It took me a long time and a substantial amount of money to get everything right, but I'm pleased with the result: the restaurants are elegant and distinctly Hawaiian. Just like the new chef's cuisine.

In the past, I've contracted a local market research firm to conduct surveys, asking guests to rate the Kahana's restaurants' ambiance on a scale of one to five.

Historically, the percentage of people that rated ambiance the top score of 5 gave me a good idea of how well we were doing. That percentage has been very high: 72%.

I've collected this year's data for you. Can you figure out if my guests are happier with my restaurants' ambiance?

Hypothesis Tests for Single Population Proportions

Alice tells you that testing Leo's claim about a proportion will be very similar to testing a mean.

Often the summary statistic we want to make a claim about is a proportion. How do we test a hypothesis about a population proportion instead of a population mean?

We know from our work with confidence intervals that the processes for estimating population

proportions and population means are virtually identical. Similarly, hypothesis tests for proportions are much like hypothesis tests for means.

Because we are examining a population proportion instead of a population mean, we use slightly different notation: we use a lower case p to represent the population proportion in place of ! for a population mean. We construct a hypothesis test to test a claim about the value of p.

Again, we formulate null and alternative hypotheses. Based on conventional wisdom or past experience, we have an initial understanding of the population proportion. The null hypothesis for a proportion test states the initial understanding. For example, in a two-sided test, the null hypothesis asserts that the population proportion, p, is equal to the initial value we had in mind.

The alternative hypothesis is the claim we are using the hypothesis test to substantiate. The alternative hypothesis typically states the opposite of the null hypothesis: it states that our initial understanding is incorrect.

As with population means, we collect a random sample and calculate the sample proportion, "p bar." However, for a hypothesis test about a population proportion, we don't need to calculate a standard deviation from the sample.

Statistical theory tells us that ", the standard deviation of the population proportion, is the square root of [p*(1 - p)]. Since we always start the test assuming the null hypothesis is true, we will calculate " using the null hypothesis proportion.

Analogously to population mean tests, we create a range of likely sample proportions around the null hypothesis proportion. To create the range, we substitute for ", the standard deviation of the underlying null hypothesis population.

If our sample proportion falls outside the range of likely sample proportions, we reject the null hypothesis. Otherwise, we cannot reject the null hypothesis.

Summary

In a hypothesis test for population proportions, we assume that the null hypothesis is true. Then, we construct a range of likely sample proportions around the null hypothesis proportion. If the sample proportion we collect falls in the rejection region, we reject the null hypothesis. Otherwise, we cannot reject the null hypothesis.

Solving the Restaurant Ambiance Problem

Once you understand hypothesis testing for means, using the same techniques on proportions is easy. By now, you're familiar with the concept of testing a hypothesis. You recognize that Leo's restaurant ambiance problem calls for a hypothesis test for a population proportion.

Leo wants you to find out if the proportion of his guests that rate restaurant ambiance "excellent" has increased. Historically, that population proportion has been 0.72. Since Leo wants to see if there has been positive change, you do a one-sided test.

The appropriate pair of hypotheses is:

a. Null hypothesis p = 0.72, alternative hypothesis: p ` 0.72

This is not the best answer. This would be a good pair of hypotheses for a two-sided test.

This is not the best answer. This would be a good pair of hypotheses if we wanted to know if the population porportion had gone down.

c. Null hypothesis p $ 0.72, alternative hypothesis: p > 0.72 This is the correct answer.

You are doing a one-sided test to see if the proportion of guests rating the restaurant "excellent" has increased. The alternative hypothesis states that the proportion has increased, and the null hypothesis states that it has not increased.

You look at Leo's data. The sample proportion is 0.81 and the sample size is 126. But what about the standard deviation?

a. You have enough information to calculate the standard deviation.

This is the correct answer. For proportions, you can calculate the standard deviation using the null hypothesis proportion.

b. You should call Leo and ask him to calculate the standard deviation for you.

Leo's phone is busy. Perhaps you should reconsider the data, and see if you can figure out the standard deviation with the information you have.

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Utility for Single Populations

Here's how you find the standard deviation for a proportion problem:

Using the appropriate formula, you calculate the standard deviation to be 0.45.

Leo wanted you to use a 95% confidence level. Now you're ready to construct a range of likely sample means around the null hypothesis value of the population proportion: 0.72.

Find the range of likely sample proportions around the null hypothesis proportion, and formulate a short answer for Leo.

a. There is not enough evidence to show that the proportion of guests rating the restaurant ambiance "excellent" has increased.

This is not the correct answer. Check your calculations: the sample proportion falls well outside the range of likely sample proportions.

b. The evidence supports Leo's claim that the proporation of guests rating the restaurant ambiance "excellent" has increased.

This is the correct answer.

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Utility for Single Populations

A one-sided test calls for a one-sided range of likely sample proportions. You need to find the upper bound for this range such that the range captures the lower 95% of the sample proportions.

The z-value for a one-sided 95% confidence level is 1.645. Substitute the null hypothesis proportion, 0.72, for p. The upper bound for the range containing the lower 95% of all sample means is 0.78. Since the sample proportion 0.81 falls in the rejection region, you reject the null hypothesis. The data provide sufficient evidence that the population proportion has, in fact, changed.

Alice presents your findings to Leo, telling him that with 95% confidence, the data you collected indicate that the difference between the historical population proportion and the proportion of the random sample is not due to chance.

Just what I wanted to hear! Thanks, you two. Exercise 1: The Ventura Insurance Company

Luther Lenya, the new product guru of The Ventura Automotive Insurance Company, is considering marketing a special insurance package to members of certain professional groups.

In particular, Luther wants to create a special package for health professionals.

To find out what rate to charge for this package, Luther conducts a preliminary study to see if health professionals are less likely to be involved in car accidents than the rest of his customer base. If the data indicate that health professionals are less likely to be involved in car accidents, then Ventura can offer health professionals a lower, more competitive rate.

In the past 5 years, 8.3% of Ventura's customers have been involved in accidents. Which of the following is the correct pair of hypotheses for solving Luther's problem?

a. Null hypothesis p = 8.3%; Alternative hypothesis p % 8.3%

This is not the best answer. These would be the appropriate hypotheses for a two-sided test.

b. Null hypothesis p $ 8.3%; Alternative hypothesis p > 8.3%

This is not the best answer. Luther wants to know if medical professionals are better drivers. The alternative hypothesis should state that medical professionals are less likely to be in accidents.

c. Null hypothesis p " 8.3%; Alternative hypothesis p < 8.3%

This is the correct answer. Luther wants a one-sided test, because he wants to know if medical professionals are better drivers. The alternative hypothesis should state that medical professionals are less likely to be in

accidents.

A sample of 240 customers in the health profession reveals that 12 (5.0%) have had accidents. If he uses a 95% confidence level, which of the following is the best conclusion Luther can come to?

a. Health professionals should be charged more for car insurance.

This is not the best answer. The sample proportion falls outside the range of likely sample proprotions around the null hypothesis proportion, and the null hypothesis should be rejected. Based on our test, we shouldn't charge health professionals higher insurance rates. Health care professionals have fewer accidents, this, if we change their rates at all, they should be charged lower rates.

b. The evidence suggests that health professionals are less likely to be involved with car accidents. This is the best answer. The range of likely sample proportions around the null hypothesis proportion does not contain the sample proportion, so we can reject the null hypothesis. With 95% confidence, the proportion of health professionals involved in car accidents is lower than the proportion of Ventura's population of drivers. c. The data provide no evidence suggests that health professionals are more or less likely to be

involved in car accidents.

This is not the best answer. The sample proportion falls outside the range of likely sample proprotions around the null hypothesis proportion, so you should reject the null hypothesis.

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Utility for Single Populations

You need to find a range of likely sample proportions. To find this range, you calculate a standard deviation. The standard deviation is 0.28.

For a one-sided test, a confidence level of 95% corresponds to a z-value of 1.645. The lower bound of this range is 0.054 = 5.4%.

With 95% confidence, the proportion of health professionals involved in car accidents is lower than the proportion of the overall population of drivers.

P-Values

After sleeping over your analysis of restaurant operations, Leo seems unsatisfied. Leo Demands a Deeper Understanding

Don't get me wrong, I appreciate your hard work. But look here: these hypothesis tests result in a "reject/don't reject" decision. If I understand you correctly, it doesn't matter how close to the border of the rejection region our sample statistic falls: "reject" is "reject."

But can't you tell me more? I want to know how strong the evidence against the null hypothesis is, not just if it is strong enough.

I'm glad you brought that issue up, Leo. We have a second method of doing hypothesis tests, one that provides a measure of the strength of the evidence.

P-Values

The evening before, Alice had acquainted you with p-values: "We can use the p-value method of hypothesis testing to make 'reject/not reject' decisions in the same way we have been doing all along. But the p-value also measures the strength of evidence against a null hypothesis."

In hypothesis tests we've done so far, we first chose the confidence level of the test. The confidence level tells us the significance level of the test, which is simply 1 minus the confidence level.

Typically, we chose a 5% significance level — a 95% confidence level — as our threshold value for rejection. Assuming that the null hypothesis is true, we reasoned that certain sample mean values are less likely to appear than others. If the mean of the sample we collected was sufficiently unlikely to appear (that is, less than 5% likely) we considered the null hypothesis implausible and rejected it. Now, rather than simply checking whether the likelihood of collecting our sample is above or below our chosen threshold, we'll ask: if the null hypothesis is true, how likely is it to choose a sample with a