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Chapter 9: Problem 33

Check that the conditions for carrying out a one-sample z test for the population proportion p are met. Lefties Simon reads a newspaper report claiming that 12% of all adults in the United States are left- handed. He wonders if 12% of the students at his large public high school are left-handed. Simon chooses an SRS of 100 students and records whether each student is right- or left-handed.

Short Answer

Expert verified
All conditions for a one-sample z test are satisfied.

Step by step solution

01

Random Sample

Simon has taken a Simple Random Sample (SRS) of 100 students from his high school. This ensures that each student had an equal chance of being selected, which is a key condition for conducting a z test.
02

Sample Size

The sample size is 100, which is large enough to justify using a normal approximation. Typically, a sample size of at least 30 is considered large enough for the central limit theorem to hold, ensuring the distribution of sample proportions is approximately normal.
03

Expected Number of Successes and Failures

For a one-sample z test, the expected number of successes should be at least 10, and the expected number of failures should also be at least 10. Successes here refer to the number of left-handed students. Calculating these: \[ np = 100 \times 0.12 = 12 \] \[ n(1 - p) = 100 \times 0.88 = 88 \] Both values are greater than 10, meeting the condition.
04

Independence

The sample size should not be more than 10% of the population to ensure independence is maintained. 100 students are likely less than 10% of the total student population at a large high school, meeting this condition.

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Key Concepts

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

Random Sample
In the context of statistics, a random sample is a subgroup of a population chosen in such a way that every individual has an equal chance of being selected. Random sampling helps to ensure that the sample represents the population without bias.

In Simon's case, he used a Simple Random Sample (SRS) methodology to select 100 students from his high school. This means that each student, regardless of any personal characteristics, had an equal opportunity to be selected for the sample.

This randomness is crucial for the validity of statistical tests like the one-sample z test, as it reduces the potential for bias and helps make the findings applicable to the larger population.
Sample Size
Sample size is an important concept in statistical testing. It refers to the number of observations or individuals in a sample. In statistical terms, Simon's sample size was 100 students. This is significant because:
  • It is large enough for the Central Limit Theorem (CLT) to apply, which states that with a sufficient sample size, the sampling distribution of the sample mean will be approximately normal.
  • It meets the requirements for using a normal approximation for proportions, which generally requires a sample size of at least 30.
  • Having a sufficient sample size helps ensure that the test results are reliable and minimizes sampling error.
This large sample size thus supports Simon's one-sample z test's validity by providing accurate estimates of the population parameters.
Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental statistical principle. It states that the distribution of the sample mean is approximately normal for a large enough sample size, regardless of the population's distribution.

In Simon's test, the CLT allows him to use a normal distribution to model the proportion of left-handed students, even if the true distribution of handedness in his school is not normal.

Key points:
  • The CLT applies here because Simon's sample size of 100 is sufficiently large.
  • It justifies using the one-sample z test for proportion, as the sample size allows the test's assumptions to be met.
CLT is essential for making inferences about population parameters when only a sample is available.
Population Proportion
Population proportion refers to the fraction of the population that possesses a particular characteristic. In the context of Simon's study, it is the proportion of left-handed students within his high school.

The newspaper report suggests a population proportion of 12% for left-handed individuals in the general adult population. Simon wonders if the same proportion applies to his school.
  • To test this, Simon uses the one-sample z test for population proportion.
  • This test helps determine whether the observed proportion in Simon鈥檚 sample significantly differs from the reported 12%.
Understanding population proportion is critical in deciding if there's a significant deviation in specific subgroups compared to a known or hypothesized population parameter.

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Most popular questions from this chapter

After once again losing a football game to the archrival, a college鈥檚 alumni association conducted a survey to see if alumni were in favor of firing the coach. An SRS of 100 alumni from the population of all living alumni was taken, and 64 of the alumni in the sample were in favor of firing the coach. Suppose you wish to see if a majority of living alumni are in favor of firing the coach. The appropriate test statistic is (a) \(z=\frac{0.64-0.5}{\sqrt{\frac{0.64(0.36)}{100}}}\) (d) \(z=\frac{0.64-0.5}{\sqrt{\frac{0.64(0.36)}{64}}}\) (b) \(t=\frac{0.64-0.5}{\sqrt{\frac{0.64(0.36)}{100}}} \quad\) (e) \(z=\frac{0.5-0.64}{\sqrt{\frac{0.5(0.5)}{100}}}\) (c) \(z=\frac{0.64-0.5}{\sqrt{\frac{0.5(0.5)}{100}}}\)

A researcher plans to conduct a significance test at the \(\alpha=0.01\) significance level. She designs her study to have a power of 0.90 at a particular alternative value of the parameter of interest. The probability that the researcher will commit a Type II error for the particular alternative value of the parameter at which she computed the power is (a) 0.01. (b) 0.10. (c) 0.89. (d) 0.90. (e) 0.99.

Explain why we aren鈥檛 safe carrying out a one-sample z test for the population proportion p. No test You toss a coin 10 times to test the hypothesis \(H_{0} : p=0.5\) that the coin is balanced.

Sweetening colas Cola makers test new recipes for loss of sweetness during storage. Trained tasters rate the sweetness before and after storage. From experience, the population distribution of sweetness losses will be close to Normal. Here are the sweetness losses (sweetness before storage minus sweetness after storage) found by tasters from a random sample of 10 batches of a new cola recipe: \(2.0 \quad 0.4 \quad 0.7 \quad 2.0 \quad-0.4 \quad 2.2 \quad-1.3 \quad 1.2 \quad 1.1 \quad 2.3\) Are these data good evidence that the cola lost sweetness? Carry out a test to help you answer this question.

After checking that conditions are met, you perform a significance test of \(H_{0} : \mu=1\) versus \(H_{a} : \mu \neq 1 .\) You obtain a P-value of 0.022 . Which of the following is true? (a) A 95\(\%\) confidence interval for \(\mu\) will include the value \(1 .\) (b) A 95\(\%\) confidence interval for \(\mu\) will include the value \(0 .\) (c) A 999\(\%\) confidence interval for \(\mu\) will include the value \(1 .\) (d) A 99\(\%\) confidence interval for \(\mu\) will include the value \(0 .\) (e) None of these is necessarily true.
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