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Chapter 4: Problem 4

A situation is described for a statistical test and some hypothetical sample results are given. In each case: (a) State which of the possible sample results provides the most evidence for the claim. (b) State which (if any) of the possible results provide no evidence for the claim. Testing to see if there is evidence that the proportion of US citizens who can name the capital city of Canada is greater than \(0.75 .\) Use the following possible sample results: Sample A: 31 successes out of 40 Sample B: \(\quad 34\) successes out of 40 Sample C: 27 successes out of 40 Sample \(\mathrm{D}: \quad 38\) successes out of 40

Short Answer

Expert verified
The sample with the most evidence for the claim is sample D, while sample C provides no evidence for the claim.

Step by step solution

01

Calculate the Proportion

The proportion is simply the division of 'successes' by the total. So, for each of the given samples, compute the proportion as follows: \n\nSample A: \(\frac{31}{40} = 0.775\) \n\nSample B: \(\frac{34}{40} = 0.85\) \n\nSample C: \(\frac{27}{40} = 0.675\) \n\nSample D: \(\frac{38}{40} = 0.95\)
02

Identify the Sample with the Most Evidence

The claim is that the proportion of US citizens who can name the capital city of Canada is greater than 0.75. So, the closer the proportion to 1, the stronger the evidence for the claim. Comparing those values, Sample D with a proportion of 0.95 gives the most evidence.
03

Identify the Sample with the Least Evidence

Now, inspecting which sample provides no evidence or contradict the claim, it's noticed that only Sample C has a proportion less than 0.75. It means all other sample results provide some evidence supporting the claim. However, Sample C with 0.675 contradicts the claim, thus it would provide no evidence in support of the claim.

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Key Concepts

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

Statistical Evidence
Statistical evidence refers to the use of sample data to assess a claim or hypothesis about a population parameter. In this problem, we are checking if the proportion of US citizens who know the capital of Canada exceeds 0.75. Statistical evidence helps determine how adequately the sample information supports the hypothesis. It involves comparing the sample results to the hypothesized value.
Understanding statistical evidence encompasses:
  • Evaluating sample proportions against the threshold (here, 0.75).
  • Interpreting how close these proportions are to 1 or exceed the threshold.
The juicier the figure extracted from sample analysis towards 1, the tougher the evidence, as it strengthens the argument favoring the claim. If the computed value is less than or very close to the hypothesized parameter, it reflects weak or insufficient statistical evidence.
Proportion Calculations
Proportion calculations form a fundamental part of analyzing sample data. In this exercise, it requires simple arithmetic 鈥� dividing the number of successful outcomes by the total number of trials.
Consider the calculation for each sample:
  • Sample A: Proportion = \( \frac{31}{40} = 0.775 \)
  • Sample B: Proportion = \( \frac{34}{40} = 0.85 \)
  • Sample C: Proportion = \( \frac{27}{40} = 0.675 \)
  • Sample D: Proportion = \( \frac{38}{40} = 0.95 \)
These calculations determine the strength of evidence each sample provides. A proportion above 0.75 supports the hypothesis while a number below opposes it. Therefore, remember: finding proportions is your initial step in determining what the sample data signifies in terms of the hypothesis.
Sample Analysis
Sample analysis involves comparing calculated proportions to the claimed parameter. It aids in judging which samples affirm or dispute the hypothesis.
In the given exercise, each sample proportion has been calculated to establish an understanding. The analysis here seeks to determine:
  • Which sample offers the most support to the claim (Sample D with 0.95).
  • Which offers no support at all by being below 0.75 (Sample C with 0.675).
Such analysis highlights not just which samples align with the preliminary assumption but also those that do not. It clarifies the landscape of data in terms of adherence to expected statistical outcomes, allowing more informed conclusions.

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Most popular questions from this chapter

The same sample statistic is used to test a hypothesis, using different sample sizes. In each case, use StatKey or other technology to find the p-value and indicate whether the results are significant at a \(5 \%\) level. Which sample size provides the strongest evidence for the alternative hypothesis? Testing \(H_{0}: p=0.5\) vs \(H_{a}: p>0.5\) using \(\hat{p}=0.55\) with each of the following sample sizes: (a) \(\hat{p}=55 / 100=0.55\) (b) \(\hat{p}=275 / 500=0.55\) (c) \(\hat{p}=550 / 1000=0.55\)

Do iPads Help Kindergartners Learn: A Subtest The Auburn, Maine, school district conducted an early literacy experiment in the fall of 2011 . In September, half of the kindergarten classes were randomly assigned iPads (the intervention group) while the other half of the classes got them in December (the control group.) Kids were tested in September and December and the study measures the average difference in score gains between the control and intervention group. \(^{41}\) The experimenters tested whether the mean score for the intervention group was higher on the HRSIW subtest (Hearing and Recording Sounds in Words) than the mean score for the control group. (a) State the null and alternative hypotheses of the test and define any relevant parameters. (b) The p-value for the test is 0.02 . State the conclusion of the test in context. Are the results statistically significant at the \(5 \%\) level? (c) The effect size was about two points, which means the mean score for the intervention group was approximately two points higher than the mean score for the control group on this subtest. A school board member argues, "While these results might be statistically significant, they may not be practically significant." What does she mean by this in this context?

4.150 Approval from the FDA for Antidepressants The FDA (US Food and Drug Administration) is responsible for approving all new drugs sold in the US. In order to approve a new drug for use as an antidepressant, the FDA requires two results from randomized double-blind experiments showing the drug is more effective than a placebo at a \(5 \%\) level. The FDA does not put a limit on the number of times a drug company can try such experiments. Explain, using the problem of multiple tests, why the FDA might want to rethink its guidelines. 4.151 Does Massage Really Help Reduce Inflammation in Muscles? In Exercise 4.112 on page \(301,\) we learn that massage helps reduce levels of the inflammatory cytokine interleukin-6 in muscles when muscle tissue is tested 2.5 hours after massage. The results were significant at the \(5 \%\) level. However, the authors of the study actually performed 42 different tests: They tested for significance with 21 different compounds in muscles and at two different times (right after the massage and 2.5 hours after). (a) Given this new information, should we have less confidence in the one result described in the earlier exercise? Why? (b) Sixteen of the tests done by the authors involved measuring the effects of massage on muscle metabolites. None of these tests were significant. Do you think massage affects muscle metabolites? (c) Eight of the tests done by the authors (including the one described in the earlier exercise) involved measuring the effects of massage on inflammation in the muscle. Four of these tests were significant. Do you think it is safe to conclude that massage really does reduce inflammation?

By some accounts, the first formal hypothesis test to use statistics involved the claim of a lady tasting tea. \({ }^{11}\) In the 1920 's Muriel Bristol- Roach, a British biological scientist, was at a tea party where she claimed to be able to tell whether milk was poured into a cup before or after the tea. R.A. Fisher, an eminent statistician, was also attending the party. As a natural skeptic, Fisher assumed that Muriel had no ability to distinguish whether the milk or tea was poured first, and decided to test her claim. An experiment was designed in which Muriel would be presented with some cups of tea with the milk poured first, and some cups with the tea poured first. (a) In plain English (no symbols), describe the null and alternative hypotheses for this scenario. (b) Let \(p\) be the true proportion of times Muriel can guess correctly. State the null and alternative hypothesis in terms of \(p\).

Do You Own a Smartphone? A study \(^{19}\) conducted in July 2015 examines smartphone ownership by US adults. A random sample of 2001 people were surveyed, and the study shows that 688 of the 989 men own a smartphone and 671 of the 1012 women own a smartphone. We want to test whether the survey results provide evidence of a difference in the proportion owning a smartphone between men and women. (a) State the null and alternative hypotheses, and define the parameters. (b) Give the notation and value of the sample statistic. In the sample, which group has higher smartphone ownership: men or women? (c) Use StatKey or other technology to find the pvalue.
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