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Chapter 10: Problem 50

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes is the large-sample \(z\) test appropriate? a. \(H_{0}: p=0.2, n=25\) b. \(H_{0}: p=0.6, n=200\) c. \(H_{0}: p=0.9, n=100\) d. \(H_{0}: p=0.05, n=75\)

Short Answer

Expert verified
The large-sample z-test is appropriate for cases b and c, i.e. \(H_{0}: p=0.6, n=200\) and \(H_{0}: p=0.9, n=100\).

Step by step solution

01

Case a: \(H_{0}: p=0.2, n=25\)

Calculate np and n(1-p) for the first case: np = \(n*p = 25*0.2 = 5\) n(1-p) = \(25*(1-0.2) = 25*0.8 = 20\) Since np < 10, this case does not meet the requirement to use the large-sample z-test.
02

Case b: \(H_{0}: p=0.6, n=200\)

Calculate np and n(1-p) for the second case: np = \(n*p = 200*0.6 = 120\) n(1-p) = \(200*(1-0.6) = 200*0.4 = 80\) In this case, both np and n(1-p) are greater than or equal to 10, so the use of a large-sample z-test is appropriate.
03

Case c: \(H_{0}: p=0.9, n=100\)

Calculate np and n(1-p) for the third case: np = \(n*p = 100*0.9 = 90\) n(1-p) = \(100*(1-0.9) = 100*0.1 = 10\) Both np and n(1-p) are greater than or equal to 10, so the use of a large-sample z-test is appropriate in this case as well.
04

Case d: \(H_{0}: p=0.05, n=75\)

Calculate np and n(1-p) for the fourth case: np = \(n*p = 75*0.05 = 3.75\) n(1-p) = \(75*(1-0.05) = 75*0.95 = 71.25\) Since np < 10, using a large-sample z-test is not appropriate for this case.
05

Conclusion

Based on the calculations, the large-sample z-test is appropriate for cases b and c, i.e. \(H_{0}: p=0.6, n=200\) and \(H_{0}: p=0.9, n=100\).

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Key Concepts

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

Large-Sample Z-Test
The Large-Sample Z-Test is a statistical method used to determine if there is a significant difference between the observed sample proportion and the hypothesized population proportion under the null hypothesis. It is specifically applicable to scenarios where the sample size is large enough for the normal approximation to the binomial distribution to be effective.
The basic requirement for applying a Large-Sample Z-Test is that both np and n(1-p) must be greater than or equal to 10. This ensures that the sample size is sufficiently large for the z-test to be valid.
  • If either np or n(1-p) is less than 10, it implies that the sample is too small for the normal approximation to hold true, making the z-test inappropriate.
  • The z-test involves calculating a z-score, which helps in determining the probability of observing a sample proportion as extreme as, or more extreme than, the one observed, under the assumption that the null hypothesis is true.
Null Hypothesis
The Null Hypothesis ( H_{0} ) is a critical concept in hypothesis testing. It provides a statement that there is no effect or no difference, and it serves as a starting point for testing statistical claims. In the context of proportion testing, the null hypothesis generally proposes a specific value for the population proportion.
For example, H_{0}: p = 0.2 suggests that the true proportion of the population is 0.2.
  • The goal of hypothesis testing is either to reject the null hypothesis or fail to reject it based on the sample data. Rejection indicates that there is sufficient evidence to support that the true population proportion is different from the one stated in H_{0} .
  • It is important to note that failing to reject the null hypothesis does not prove that H_{0} is true; it merely indicates a lack of strong evidence against it.
Sample Size
Sample size is a fundamental element in hypothesis testing, greatly impacting the reliability and validity of test results. In proportion testing, it is crucial because it affects the calculations of np and n(1-p), which are determinants for the applicability of the Large-Sample Z-Test.
A larger sample size increases the likelihood that the test results will truly reflect the actual population characteristics.
  • For the Large-Sample Z-Test to be valid, both np and n(1-p) should be at least 10. This threshold ensures that the sample is large enough for the binomial distribution to approximate a normal distribution effectively.
  • Inadequate sample sizes can lead to inaccurate results because they fail to capture the population's variability sufficiently, leading to potential misinterpretation.
Proportion Testing
Proportion Testing is a statistical method used to compare an observed sample proportion to a known population proportion to determine if there is a significant difference. It often involves the use of a Z-Test when conditions allow.
  • The test starts with the formulation of the null hypothesis stating a specific population proportion, such as H_{0}: p = 0.6 , which implies that 60% of the population has the characteristic of interest.
  • The objective is to test this hypothesis using sample data to determine if there is a significant departure from the null hypothesis proportion.
  • Proportion testing is widely used in areas like public health, quality control, and market research for making informed decisions based on sample data.
Testing for proportions involves verifying the assumptions of the Large-Sample Z-Test and understanding the context of the problem to ensure that the test is applicable and the results meaningful.

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Most popular questions from this chapter

According to a survey of a random sample of 2278 adult Americans conducted by the Harris Poll ("Do Americans Prefer Name Brands or Store Brands? Well, That Depends" (theharrispoll.com, February 11, 2015, retrieved November 29,2016 ), 1162 of those surveyed said that they prefer name brands to store brands when purchasing frozen vegetables. Suppose that you want to use this information to determine if there is convincing evidence that a majority of adult Americans prefer name-brand frozen vegetables over store brand frozen vegetables. a. What hypotheses should be tested in order to answer this question? b. The \(P\) -value for this test is 0.173 . What conclusion would you reach if \(\alpha=0.05 ?\)

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes is the large-sample \(z\) test appropriate? a. \(H_{0}: p=0.8, n=40\) b. \(H_{0}: p=0.4, n=100\) c. \(H_{0}: p=0.1, n=50\) d. \(H_{0}: p=0.05, n=750\)

The article "Most Customers OK with New Bulbs" (USA TODAY, February 18,2011 ) describes a survey of 1016 randomly selected adult Americans. Each person in the sample was asked if they have replaced standard light bulbs in their home with the more energy efficient compact fluorescent (CFL) bulbs. Suppose you want to use the survey data to determine if there is evidence that more than \(70 \%\) of adult Americans have replaced standard bulbs with CFL bulbs. Let \(p\) denote the population proportion of all adult Americans who have replaced standard bulbs with CFL bulbs. a. Describe the shape, center, and variability of the sampling distribution of \(\hat{p}\) for random samples of size 1016 if the null hypothesis \(H_{0}: p=0.70\) is true. b. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.72\) for a sample of size 1016 if the null hypothesis \(H_{0}: p=0.70\) were true? Explain why or why not. c. Would you be surprised to observe a sample proportion as large as \(\hat{p}=0.75\) for a sample of size 1016 if the null hypothesis \(H_{0}: p=0.70\) were true? Explain why or why not. d. The actual sample proportion observed in the study was \(\hat{p}=0.71 .\) Based on this sample proportion, is there convincing evidence that more than \(70 \%\) have replaced standard bulbs with CFL bulbs, or is this sample proportion consistent with what you would expect to see when the null hypothesis is true? Support your answer with a probability calculation.

"Most Like It Hot" is the title of a press release issued by the Pew Research Center (March 18, 2009, www.pewsocialtrends. org). The press release states that "by an overwhelming margin, Americans want to live in a sunny place." This statement is based on data from a nationally representative sample of 2260 adult Americans. Of those surveyed, 1288 indicated that they would prefer to live in a hot climate rather than a cold climate. Suppose that you want to determine if there is convincing evidence that a majority of all adult Americans prefer a hot climate over a cold climate. a. What hypotheses should be tested in order to answer this question? b. The \(P\) -value for this test is 0.000001 . What conclusion would you reach if \(\alpha=0.01 ?\)

Which of the following specify legitimate pairs of null and alternative hypotheses? a. \(H_{0}: p=0.25 \quad H_{i}: p>0.25\) b. \(H_{0}: p<0.40 \quad H_{i}: p>0.40\) c. \(H_{0}: p=0.40 \quad H: p<0.65\) d. \(H_{0}: p \neq 0.50 \quad H_{i}: p=0.50\) e. \(H_{\mathrm{n}}: p=0.50 \quad H_{i}: p>0.50\) f. \(H_{0}: \hat{p}=0.25 \quad H_{e^{\prime}} \hat{p}>0.25\)
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