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Chapter 19: Problem 17

Yahoo surveyed 2400 U.S. men. 1224 of the men identified themselves as the primary grocery shopper in their household. a. Estimate the percentage of all American males who identify themselves as the primary grocery shopper. Use a \(98 \%\) confidence interval. Check the conditions first. b. A grocery store owner believed that only \(45 \%\) of men are the primary grocery shopper for their family, and targets his advertising accordingly. He wishes to conduct a hypothesis test to see if the fraction is in fact higher than \(45 \% .\) What does your confidence interval indicate? c. What is the level of significance of this test? Explain.

Short Answer

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a. The estimated percentage is about 51% with a 98% confidence interval. b. The fractional value proposed by the grocery store owner, 0.45, is less than our confidence interval, indicating the fraction could be higher than this value. c. The level of significance is 2%, meaning there is a 2% chance of incorrectly concluding the fraction of men who are the primary shoppers is greater than 45%, if it was actually 45%.

Step by step solution

01

Identify the sample proportion

First, find the sample proportion \(p\). This is the ratio of men who identified themselves as the primary grocery shopper in their household to the total number of men surveyed. Calculate \(p\) by dividing 1224 by 2400 to get approximately 0.51.
02

Check the conditions for constructing a confidence interval

The two primary conditions required for creating a confidence interval are: 1) the sample must be random, and 2) the sample must be large enough, specifically \(np > 5\) and \(n(1-p) > 5\). The problem doesn't specify whether the sample is random, but since it's a survey, we'll typically make that assumption. As for condition 2, with \(n=2400\) and \(p=0.51\), both \(np\) and \(n(1-p)\) are significantly greater than 5, so condition 2 is met.
03

Calculate the confidence interval

The confidence interval equation is \[ p \pm Z \sqrt{ (p(1-p)/n) } \]. In this case, the Z-value for a 98% confidence interval is approximately 2.33 (you can find this in a standard Z-score table). Substituting into the formula gives us a confidence interval of \[0.51 \pm 2.33 \sqrt{ (0.51(1-0.51)/2400) }\]
04

Compare the confidence interval with the value stated by the grocery store owner

Part b asks if the fraction of primary shoppers is possibly higher than 45%. We can simply compare 0.45 to our 98% confidence interval. If 0.45 is not in the interval, then the fraction could indeed be higher than 45%.
05

Identify the level of significance

The level of significance for a test is generally expressed as a probability value. It represents the probability of rejecting the null hypothesis when it is in fact true. For a 98% confidence interval, the level of significance is 100% - 98% = 2%. This means there is a 2% chance of incorrectly concluding that the fraction of men who are the primary shoppers is greater than 45%, if it was actually 45%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing
Hypothesis testing is a statistical method used to make inferences about population parameters based on a sample statistic. In the context of this exercise, the grocery store owner wishes to test the hypothesis that more than 45% of men are primary grocery shoppers.
  • To start a hypothesis test, a null hypothesis (ull_hypothesis) is proposed. It generally reflects the current belief or the status quo, which in this case is that 45% of men are primary grocery shoppers.
  • The alternative hypothesis (ull_hypothesis_a) is the statement that the researcher wishes to prove; here, it's that more than 45% of men are primary shoppers.
  • The test statistic鈥攊n this case, the sample proportion鈥攊s then used to calculate a p-value or to compare against a predetermined critical value, allowing the researcher to assess the null hypothesis's plausibility.
  • Lastly, the researcher decides to reject or not reject the null hypothesis based on the level of significance and the calculated test statistic.

For the grocery store owner, the hypothesis test might reveal if his advertising strategy should shift to target a larger percentage of male primary shoppers.
Sample Proportion
The sample proportion (ull_p) is the estimated probability of an event occurring within the sample. It is calculated by dividing the number of favorable cases by the total number of cases in the sample.
  • In our example, the favorable cases are the men who identified themselves as primary grocery shoppers, and the total number of cases is the total number of men surveyed.
  • The sample proportion here, which is approximately 0.51, represents the proportion of the sampled men who are primary grocery shoppers.
  • It is crucial to ensure that the sample is representative of the population so that the sample proportion can serve as a reliable estimate of the population proportion.

This concept is foundational as it serves as the core metric from which we infer population parameters.
Level of Significance
The level of significance denoted by ull_alpha, is the threshold for determining when a result is statistically significant enough to reject the null hypothesis.
  • It is often set before data collection, with common values being 0.05, 0.01, or 0.10, representing a 5%, 1%, or 10% chance of rejecting the null hypothesis when it is actually true (also known as a Type I error).
  • In the context of the given exercise, the level of significance is 2%, derived from the fact that the confidence interval is 98%.
  • This signifies that the grocery store owner would be taking a 2% risk of concluding that a higher proportion of men are primary shoppers when the proportion is actually 45% or less.

The level of significance acts as a gatekeeper for determining the strength of the evidence against the null hypothesis.
Z-score
A Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean.
  • If the Z-score is 0, it indicates that the data point's score is identical to the mean score.
  • A Z-score of 1.0 would indicate a value that is one standard deviation from the mean, while a Z-score of -2.5 would indicate a value that is 2.5 standard deviations below the mean.
  • In confidence interval calculations, the Z-score helps determine the margin of error. For a 98% confidence level, the Z-score is approximately 2.33, which was used in the exercise to determine the upper and lower bounds of the interval where the true population proportion is likely to fall.

Understanding Z-scores is fundamental to interpreting the relationship between sample data and the population mean.
Null Hypothesis
The null hypothesis (ull_hypothesis_0) is a statement of no effect or no difference used as a baseline to compare against evidence in the data.
  • It is the hypothesis that researchers aim to test against and possibly reject in favor of an alternative hypothesis (ull_hypothesis_a).
  • In this scenario, the null hypothesis might state that the population proportion of primary male shoppers is 0.45 or less.
  • The evidence that would lead to rejecting this null hypothesis comes from the sample proportion and the accompanying confidence interval computed from the survey data.

Understanding the null hypothesis is critical in hypothesis testing as it provides a starting point for statistical significance evaluation.

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Most popular questions from this chapter

For each of the following situations, state whether a Type I, a Type II, or neither error has been made. a. A test of \(\mathrm{H}_{0}: \mu=25\) vs. \(\mathrm{H}_{\mathrm{A}}: \mu>25\) rejects the null hypothesis. Later it is discovered that \(\mu=24.9\). b. A test of \(\mathrm{H}_{0}: p=0.8\) vs. \(\mathrm{H}_{\mathrm{A}}: p<0.8\) fails to reject the null hypothesis. Later it is discovered that \(p=0.9\). c. A test of \(\mathrm{H}_{0}: p=0.5\) vs. \(\mathrm{H}_{\mathrm{A}}: p \neq 0.5\) rejects the null hypothesis. Later it is discovered that \(p=0.65\). d. A test of \(\mathrm{H}_{0}: p=0.7\) vs. \(\mathrm{H}_{\mathrm{A}}: p<0.7\) fails to reject the null hypothesis. Later it is discovered that \(p=0.6\).

Which of the following are true? If false, explain briefly. a. AP-value of 0.01 means that the null hypothesis is false. b. AP-value of 0.01 means that the null hypothesis has a 0.01 chance of being true. c. AP-value of 0.01 is evidence against the null hypothesis. d. AP-value of 0.01 means we should definitely reject the null hypothesis.

Which of the following are true? If false, explain briefly. a. If the null hypothesis is true, you'll get a high P-value. b. If the null hypothesis is true, a P-value of 0.01 will occur about \(1 \%\) of the time. c. A P-value of 0.90 means that the null hypothesis has a good chance of being true. d. AP-value of 0.90 is strong evidence that the null hypothesis is true.

In a drawer are two coins. They look the same, but one coin produces heads \(90 \%\) of the time when spun while the other one produces heads only \(30 \%\) of the time. You select one of the coins. You are allowed to spin it once and then must decide whether the coin is the \(90 \%\) - or the \(30 \%\) -head coin. Your null hypothesis is that your coin produces \(90 \%\) heads. a. What is the alternative hypothesis? b. Given that the outcome of your spin is tails, what would you decide? What if it were heads? c. How large is \(\alpha\) in this case? d. How large is the power of this test? (Hint: How many possibilities are in the alternative hypothesis?) e. How could you lower the probability of a Type I error and increase the power of the test at the same time?

Before lending someone money, banks must decide whether they believe the applicant will repay the loan. One strategy used is a point system. Loan officers assess information about the applicant, totaling points they award for the person's income level, credit history, current debt burden, and so on. The higher the point total, the more convinced the bank is that it's safe to make the loan. Any applicant with a lower point total than a certain cutoff score is denied a loan. We can think of this decision as a hypothesis test. Since the bank makes its profit from the interest collected on repaid loans, their null hypothesis is that the applicant will repay the loan and therefore should get the money. Only if the person's score falls below the minimum cutoff will the bank reject the null and deny the loan. This system is reasonably reliable, but, of course, sometimes there are mistakes. a. When a person defaults on a loan, which type of error did the bank make? b. Which kind of error is it when the bank misses an opportunity to make a loan to someone who would have repaid it? c. Suppose the bank decides to lower the cutoff score from 250 points to 200 . Is that analogous to choosing a higher or lower value of \(a\) for a hypothesis test? Explain. d. What impact does this change in the cutoff value have on the chance of each type of error?
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