Background:

For this activity, we will use example 1. Here is a summary of what we have found:

The results of this study—64 defective products out of 400—were statistically significant in the sense that they provided enough evidence to conclude that the repair indeed reduced the proportion of defective products from 0.20 (the proportion prior to the repair).

Even though the results—a sample proportion of defective products of 0.16—are statistically significant, it is not clear whether the results indicate that the repair was effective enough to meet the company’s needs, or, in other words, whether these results have a practical importance.

If the company expected the repair to eliminate defective products almost entirely, then even though statistically, the results indicate a significant reduction in the proportion of defective products, this reduction has very little practical importance, because the repair was not effective in achieving what it was supposed to.

To make sure you understand this important distinction between statistical significance and practical importance, we will push this a bit further.

Consider the same example, but suppose that when the company examined the 400 randomly selected products, they found that 78 of them were defective (instead of 64 in the original problem):

Consider now another variation on the same problem. Assume now that over a period of a month following the repair, the company randomly selected 20,000 products, and found that 3,900 of them were defective.

Note that the sample proportion of defective products is the same as before , 0.195, which as we established before, does not indicate any practically important reduction in the proportion of defective products.

**Summary:** This is perhaps an “extreme” example, yet it is effective in illustrating the important distinction between practical importance and statistical significance. A reduction of 0.005 (or 0.5%) in the proportion of defective products probably does not carry any practical importance, however, because of the large sample size, this reduction is statistically significant. In general, with a sufficiently large sample size you can make any result that has very little practical importance statistically significant. This suggests that when interpreting the results of a test, you should always think not only about the statistical significance of the results but also about their practical importance.

This document is linked from More about Hypothesis Testing.

]]>Ho: p = .40

Ha: p > .40

The results are reported to be not statistically significant, with a p-value of 0.214.

Decide whether each of the following statements is a valid conclusion or an invalid conclusion, based on the study:

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]]>**Background: **This activity is based on the results of a recent study on the safety of airplane drinking water that was conducted by the U.S. Environmental Protection Agency (EPA). A study found that out of a random sample of 316 airplanes tested, 40 had coliform bacteria in the drinking water drawn from restrooms and kitchens. As a benchmark comparison, in 2003 the EPA found that about 3.5% of the U.S. population have coliform bacteria-infected drinking water. The question of interest is whether, based on the results of this study, we can conclude that drinking water on airplanes is more contaminated than drinking water in general.

Here is an example of possible output from software for this problem. We will verify the results ourselves.

This document is linked from Proportions (Step 4 & Summary).

]]>In this activity we will use example 2. Recall that we’ve just completed this example, and summarized it using the following figure:

We’ve seen that the evidence the data provided—19 marijuana users out of a sample of 100—was not enough for us to conclude that the proportion of marijuana users in the college is higher than the national figure (.157). An interesting question, therefore is: how many marijuana users out of 100 should we have found for it to be enough evidence to reject Ho? 19 was not enough, but what would have been enough? 21? 25?

To help us answer this question more accurately, lets look at a table that lists various sample counts/proportions of users, the corresponding z statistic, and the associated p-value.

Note that we highlighted in red the result we found in our sample—19 users.

This is a great opportunity to see how the p-value “works” as a measure of evidence against Ho; the smaller it is the more evidence is “stored” in the data against Ho. Obviously, if finding 19 marijuana users was not enough evidence to reject Ho. We therefore conclude that p, the proportion of marijuana users in the college is higher than 0.157 (the national figure), then anything below 19 would not be enough either, since it provides even less evidence against Ho.

See in the green section of the table how this is depicted by the values of the p-value, which get larger as the number of marijuana users gets smaller. On the other hand, it is pretty clear that the more marijuana users we see in our sample, the more evidence we have to reject Ho and conclude that p > 0.157. Indeed, note that the p-values get smaller as the number of marijuana users increases.

This document is linked from Proportions (Step 4 & Summary).

]]>In 2007, a Gallup poll estimated that 45% of U.S. adults rated their financial situation as “good.” We want to know if the proportion is smaller this year. We gather a random sample of 100 U.S. adults this year and find that 39 rate their financial situation as “good.” Use the output from Minitab to complete the following statements about the p-value. Use numbers from the output to fill in the blanks.

The trustees of a local school district commission a survey to determine voter opinions about a possible bond measure to fund school upgrades. In a poll of 293 of the district’s 5,019 registered voters, 178 would support the bond measure. A hypothesis test was conducted using Minitab to determine if such a bond would pass with the required 55% of the vote.

Do zinc supplements reduce a child’s risk of catching a cold? A medical study reports a p-value of 0.03. Are the following interpretations of the p-value valid or invalid?

This document is linked from Proportions (Step 3).

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This document is linked from Proportions (Step 3).

]]>**Scenario 1:** The UCLA Internet Report (February 2003) estimated that roughly 8.7% of Internet users are extremely concerned about credit card fraud when buying online. Has that figure changed since? To test this, a random sample of 100 Internet users was chosen. When interviewed, 10 said that they were extremely worried about credit card fraud when buying online. Let p be the proportion of all Internet users who are concerned about credit card fraud.

**Scenario 2:** The UCLA Internet Report (February 2003) estimated that a proportion of roughly .75 of online homes are still using dial-up access, but claimed that the use of dial-up is declining. Is that really the case? To examine this, a follow-up study was conducted a year later in which out of a random sample of 1,308 households that had an Internet connection, 804 were connecting using a dial-up modem. Let p be the proportion of all U.S. Internet-using households that have dial-up access.

**Scenario 3:** According to the UCLA Internet Report (February 2003) the use of the Internet at home is growing steadily. The report estimated that roughly 59.3% of households in the Unites States have Internet access at home. Has that trend continued since the report was released? To study this, a random sample of 1,200 households from a big metropolitan area was chosen, and it was found that 972 had an Internet connection. Let p be the proportion of U.S. households that have Internet access.

This document is linked from Proportions (Step 2).

]]>In 2007, a Gallup poll estimated that 45% of U.S. adults rated their financial situation as “good.” Is the proportion different for this year? Which of the following samples could be used to test the null hypothesis p = 0.45? Mark each as valid (OK to use to test the hypothesis) or not valid (should not be used to test the hypothesis).

We plan to poll 200 students enrolled in statistics at your college by distributing surveys during class. Which of the following hypotheses could be tested with the survey results? Mark each as valid (OK to use to test the hypothesis) or not valid (should not be used to test the hypothesis.)

This document is linked from Proportions (Step 2).

]]>Here is the sampling distribution for the proportion of females in random samples of n students. The standard deviation is approximately 0.10. Lines indicate a distance of 1 and 2 standard deviations above and below the mean.

Here is the sampling distribution for the proportion of supporters in random samples of 25 adults. The standard deviation is approximately 0.10.

If we increase the sample size to 100, the standard deviation decreases to approximately 0.05, as shown.

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]]>**Background:**

Recall from a previous activity the results of a study on the safety of airplane drinking water that was conducted by the U.S. Environmental Protection Agency (EPA). A study found that out of a random sample of 316 airplanes tested, 40 had coliform bacteria in the drinking water drawn from restrooms and kitchens. As a benchmark comparison, in 2003 the EPA found that about 3.5% of the U.S. population have coliform bacteria-infected drinking water. The question of interest is whether, based on the results of this study, we can conclude that drinking water on airplanes is more contaminated than drinking water in general. Let p be the proportion of contaminated drinking water on airplanes.

In a previous activity we tested Ho: p = 0.035 vs. Ha: p > 0.035 and found that the data provided extremely strong evidence to reject Ho. We concluded that the proportion of contaminated drinking water in airplanes is larger than the proportion of contaminated drinking water in general (which is 0.035).

Now that we’ve concluded that, all we know about p is that we have very strong evidence that it is higher than 0.035. However, we have no sense of its magnitude. It will make sense to follow up the test by estimating p with a 95% confidence interval.

This document is linked from More about Hypothesis Testing.

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