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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

Mercury Levels in Fish Figure 4.26 shows a scatterplot of the acidity (pH) for a sample of \(n=53\) Florida lakes vs the average mercury level (ppm) found in fish taken from each lake. The full dataset is introduced in Data 2.4 on page 68 and is available in FloridaLakes. There appears to be a negative trend in the scatterplot, and we wish to test whether there is significant evidence of a negative association between \(\mathrm{pH}\) and mercury levels. (a) What are the null and alternative hypotheses? (b) For these data, a statistical software package produces the following output: $$ r=-0.575 \quad p \text { -value }=0.000017 $$ Use the p-value to give the conclusion of the test. Include an assessment of the strength of the evidence and state your result in terms of rejecting or failing to reject \(H_{0}\) and in terms of \(\mathrm{pH}\) and mercury. (c) Is this convincing evidence that low pH causes the average mercury level in fish to increase? Why or why not?

Rolling Dice You roll a die 60 times and record the sample proportion of fives, and you want to test whether the die is biased to give more fives than a fair die would ordinarily give. To find the p-value for your sample data, you create a randomization distribution of proportions of fives in many simulated samples of size 60 with a fair die. (a) State the null and alternative hypotheses. (b) Where will the center of the distribution be? Why? (c) Give an example of a sample proportion for which the number of 5 's obtained is less than what you would expect in a fair die. (d) Will your answer to part (c) lie on the left or the right of the center of the randomization distribution? (e) To find the p-value for your answer to part (c), would you look at the left, right, or both tails? (f) For your answer in part (c), can you say anything about the size of the p-value?

Exercises 4.117 to 4.122 give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.6\) vs \(H_{a}: p>0.6\) Sample data: \(\hat{p}=52 / 80=0.65\) with \(n=80\)

Pesticides and ADHD Are children with higher exposure to pesticides more likely to develop ADHD (attention-deficit/hyperactivity disorder)? In a recent study, authors measured levels of urinary dialkyl phosphate (DAP, a common pesticide) concentrations and ascertained ADHD diagnostic status (Yes/No) for 1139 children who were representative of the general US population. \(^{8}\) The subjects were divided into two groups based on high or low pesticide concentrations, and we compare the proportion with ADHD in each group. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) In the sample, children with high pesticide levels were more likely to be diagnosed with ADHD. Can we necessarily conclude that, in the population, children with high pesticide levels are more likely to be diagnosed with ADHD? (Whether or not we can make this generalization is, in fact, the statistical question of interest.) (c) To assess statistical significance, we assume the null hypothesis is true. What does that mean in this case? State your answer in terms of pesticides and ADHD. (d) The study found the results to be statistically significant. Which of the hypotheses, \(H_{0}\) or \(H_{a}\), is no longer a very plausible possibility? (e) What do the statistically significant results imply about pesticide exposure and ADHD?

Penalty Shots in Soccer A recent article noted that it may be possible to accurately predict which way a penalty-shot kicker in soccer will direct his shot. \({ }^{23}\) The study finds that certain types of body language by a soccer player-called "tells"-can be accurately read to predict whether the ball will go left or right. For a given body movement leading up to the kick, the question is whether there is strong evidence that the proportion of kicks that go right is significantly different from one-half. (a) What are the null and alternative hypotheses in this situation? (b) If sample results for one type of body movement give a p-value of \(0.3184,\) what is the conclusion of the test? Should a goalie learn to distinguish this movement? (c) If sample results for a different type of body movement give a p-value of \(0.0006,\) what is the conclusion of the test? Should a goalie learn to distinguish this movement?
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