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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 significant 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: \(\quad 31\) successes out of 40 Sample B: \(\quad 34\) successes out of 40 Sample C: \(\quad 27\) successes out of 40 Sample \(\mathrm{D}: \quad 38\) successes out of 40

Short Answer

Expert verified
Sample D (38 successes out of 40) provides the most significant support for the claim as it has the highest proportion greater than 0.75. Sample C (27 successes out of 40) provides no evidence as its proportion is lesser than 0.75.

Step by step solution

01

Calculate the Sample Proportions

The sample proportions are calculated by dividing the number of successes by the total sample size. For Sample A: \(\frac{31}{40} = 0.775\) For Sample B: \(\frac{34}{40} = 0.85\)For Sample C: \(\frac{27}{40} = 0.675\)For Sample D: \(\frac{38}{40} = 0.95\)
02

Compare Sample Proportions to the Hypothesized Proportion

The hypothesized proportion is 0.75. Compare each sample proportion to this value. - Sample A: 0.775 > 0.75- Sample B: 0.85 > 0.75- Sample C: 0.675 < 0.75- Sample D: 0.95 > 0.75
03

Determine Levels of Evidence

Samples that support the claim have proportions greater than 0.75, and samples that do not support the claim have proportions lesser than or equal to 0.75.- Sample A provides some support.- Sample B provides significant support.- Sample C provides no support.- Sample D provides the most significant support since it has the highest proportion greater than 0.75.

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Key Concepts

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

Understanding Sample Proportion
In statistics, the sample proportion is an estimate of a population proportion. It's a way to predict or infer information about a larger group based on a smaller, observed subset. Calculating the sample proportion involves dividing the number of successful outcomes by the total number of trials or observations. For instance, if we have 31 people out of 40 who correctly name the capital city of Canada, the sample proportion would be \( \frac{31}{40} = 0.775 \). This means 77.5% of the sample got it right.

This value is crucial in hypothesis testing, as it helps to determine whether or not your observed data supports a certain claim about the population. If the sample proportion is close to what you hypothesize about the entire population, the sample offers some evidence for the claim. In our exercise, we have different sample proportions for each sample (A, B, C, D), allowing us to see which offers more or less support for the hypothesis that more than 75% of US citizens can name the capital city of Canada.
The Role of Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions about the properties of a population, based on a sample. It helps determine if there is enough evidence to support a claim about a population parameter, such as the proportion of people who can name the capital of a country.

Here's how it generally works:
  • Formulate a null hypothesis \((H_0)\): This usually states that there is no effect or difference, for instance, the proportion is \(0.75\).
  • Set up an alternative hypothesis \((H_A)\): This is what you seek evidence for, such as a proportion greater than \(0.75\).
  • Calculate the sample statistic: Like the sample proportion.
  • Use a test statistic to measure the evidence: Decide how extreme the sample data are.
  • Make a decision based on a significance level: Often \(0.05\), which represents a 5% chance of committing a Type I error (rejecting a true null hypothesis).
In the provided exercise, hypothesis testing helps us see which sample offers the most significant evidence against the null hypothesis and supports the claim of higher proportions.
Comparing Proportions for Evidence
Proportion comparison involves evaluating different proportions to determine which provides stronger support for a hypothesis. This is often performed when you have multiple samples or groups, and you want to know how they stack up against each other regarding a specific claim.

In the example scenario, the hypotheses test whether more than 75% of US citizens can correctly name the capital of Canada. Each sample gives a proportion:
  • Sample A: \(0.775\)
  • Sample B: \(0.85\)
  • Sample C: \(0.675\)
  • Sample D: \(0.95\)
Comparing these to the hypothesized proportion of \(0.75\), Sample D shows the highest evidence in favor of the claim, given its large proportion well above 0.75. Samples with lower proportions like Sample C provide no support as they fall below the benchmark of \(0.75\). Determining significance involves not just looking at which proportion is greater, but which is largest compared to the standard or hypothesized value.

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Most popular questions from this chapter

Arsenic in Chicken Data 4.5 on page 228 introduces a situation in which a restaurant chain is measuring the levels of arsenic in chicken from its suppliers. The question is whether there is evidence that the mean level of arsenic is greater than 80 ppb, so we are testing \(H_{0}: \mu=80\) vs \(H_{a}: \mu>80\), where \(\mu\) represents the average level of arsenic in all chicken from a certain supplier. It takes money and time to test for arsenic so samples are often small. Suppose \(n=6\) chickens from one supplier are tested, and the levels of arsenic (in ppb) are: \(68, \quad 75\) 81, \(\quad 93\) 134 (a) What is the sample mean for the data? (b) Translate the original sample data by the appropriate amount to create a new dataset in which the null hypothesis is true. How do the sample size and standard deviation of this new dataset compare to the sample size and standard deviation of the original dataset? (c) Write the six new data values from part (b) on six cards. Sample from these cards with replacement to generate one randomization sample. (Select a card at random, record the value, put it back, select another at random, until you have a sample of size \(6,\) to match the original sample size.) List the values in the sample and give the sample mean. (d) Generate 9 more simulated samples, for a total of 10 samples for a randomization distribution. Give the sample mean in each case and create a small dotplot. Use an arrow to locate the original sample mean on your dotplot.

Indicate whether the analysis involves a statistical test. If it does involve a statistical test, state the population parameter(s) of interest and the null and alternative hypotheses. Polling 1000 people in a large community to determine if there is evidence for the claim that the percentage of people in the community living in a mobile home is greater than \(10 \%\)

Beer and Mosquitoes Does consuming beer attract mosquitoes? Exercise 4.17 on page 232 discusses an experiment done in Africa testing possible ways to reduce the spread of malaria by mosquitoes. In the experiment, 43 volunteers were randomly assigned to consume either a liter of beer or a liter of water, and the attractiveness to mosquitoes of each volunteer was measured. The experiment was designed to test whether beer consumption increases mosquito attraction. The report \(^{27}\) states that "Beer consumption, as opposed to water consumption, significantly increased the activation... of An. gambiae [mosquitoes]... \((P<0.001)\) (a) Is this convincing evidence that consuming beer is associated with higher mosquito attraction? Why or why not? (b) How strong is the evidence for the result? Explain. (c) Based on these results, is it reasonable to conclude that consuming beer causes an increase in mosquito attraction? Why or why not?

In Exercises 4.71 to \(4.74,\) using the p-value given, are the results significant at a \(10 \%\) level? At a \(5 \%\) level? At a 1\% level? $$ \text { p-value }=0.0320 $$

Guilty Verdicts in Court Cases A reporter on cnn.com stated in July 2010 that \(95 \%\) of all court cases that go to trial result in a guilty verdict. To test the accuracy of this claim, we collect a random sample of 2000 court cases that went to trial and record the proportion that resulted in a guilty verdict. (a) What is/are the relevant parameter(s)? What sample statistic(s) is/are used to conduct the test? (b) State the null and alternative hypotheses. (c) We assess evidence by considering how likely our sample results are when \(H_{0}\) is true. What does that mean in this case?
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