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Chapter 9: Problem 82

Consider the null hypothesis \(H_{0}: p=.25 .\) Suppose a random sample of 400 observations is taken to perform this test about the population proportion. Using \(\alpha=.01\), show the rejection and nonrejection regions and find the critical value(s) of \(z\) for a a. left-tailed test b. two-tailed test c. right-tailed test

Short Answer

Expert verified
For an alpha level of \(\alpha = .01\), the critical z-scores and the resulting rejection/nonrejection regions are: a) In a left-tailed test, the critical value is -2.33 (rejection region: z < -2.33, nonrejection region: z > -2.33). b) In a two-tailed test, the critical values are -2.58 and +2.58 (rejection regions: z < -2.58 and z > +2.58, nonrejection region: -2.58 < z < +2.58). c) In a right-tailed test, the critical value is +2.33 (rejection region: z > +2.33, nonrejection region: z < +2.33).

Step by step solution

01

Determine the z-score critical value for a left-tailed test

The first section of the problem asks for a left-tailed test. In a left-tailed test, the rejection region is located entirely in the left tail of the distribution. For an alpha level of \(\alpha = .01\), the critical z-score can be identified from the standard normal distribution table or a z-score calculator, yielding a critical z-score of approximately -2.33.
02

Determine the z-score critical value for a two-tailed test

The second part of the problem sets forth a two-tailed test. In this type of test, rejection regions are found in both tails of the distribution. Considering \(\alpha = .01\), this alpha level is split between the two tails, so each tail has an area of .005. Referring again to the standard normal distribution table, two z-scores correspond to the areas above and below the z-scores, amounting to \(0.5000 - 0.0050 = 0.4950\) each. These points yield critical z-scores of approximately -2.58 and +2.58.
03

Determine the z-score critical value for a right-tailed test

The third part of the problem entails a right-tailed test. For this, the rejection region is located in the right tail of the distribution. With \(\alpha = .01\), the z-score related to the alpha level in the right tail can be obtained from standard tables or a calculator, rendering a critical z-score of approximately +2.33.

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Key Concepts

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

Left-Tailed Test
A left-tailed test is a hypothesis test where the rejection region is entirely on the left side of the distribution. It is used when you want to determine if a sample proportion is significantly less than the hypothesized population proportion. The key concept here is that the test aims to see if there is a statistically significant decrease in the population parameter. To perform a left-tailed test:
  • State the null and alternative hypothesis, for instance,\( H_0: p = 0.25 \) and \( H_A: p < 0.25 \).
  • Choose a significance level, often denoted by \(\alpha\). In the exercise, \(\alpha = 0.01\).
  • Determine the critical value from the z-distribution. For \(\alpha = 0.01\), the critical z-score is approximately -2.33.

A test statistic that falls below this critical value indicates a significant result, and the null hypothesis may be rejected in favor of the alternative.
Two-Tailed Test
In a two-tailed test, the hypothesis test evaluates the possibility of a parameter falling significantly either above or below the hypothesized value. This type of test is suitable when you do not have any directionality in assessing deviations from the null hypothesis.
When executing a two-tailed test from the exercise:
  • Formulate the hypotheses appropriately. For example, \( H_0: p = 0.25 \) and \( H_A: p eq 0.25 \).
  • The significance level, \(\alpha = 0.01\), is divided between the two tails, leading to \(\alpha/2 = 0.005\) for each tail.
  • The associated critical z-scores are approximately -2.58 and +2.58.

If the test statistic lies beyond either critical z-score, the null hypothesis is rejected. This setup allows for detecting both an increase and a decrease different from the hypothesized proportion.
Right-Tailed Test
For a right-tailed test, the rejection region resides completely in the right tail of the distribution. This setup is used when the goal is to detect if the sample proportion is significantly greater than the hypothesized population proportion. Here's how it's structured:
  • State the null and alternative hypotheses, such as \( H_0: p = 0.25 \) and \( H_A: p > 0.25 \).
  • Select the significance level, \(\alpha = 0.01\) in this case.
  • The critical z-score, obtained from the z-table, is approximately +2.33 for \(\alpha = 0.01\) in the right tail.

Any test statistic that exceeds this critical value suggests significant results. Thus, the null hypothesis can be rejected, supporting that the population proportion exceeds the hypothesized value.
Z-Score
In hypothesis testing, the z-score is a standard score that allows for comparing a sample proportion to a population proportion during the test. The z-score tells you how many standard deviations away the sample proportion is from the hypothesized population proportion. Calculating the z-score:
  • The formula for the z-score for a sample proportion is:
    \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]
    where \( \hat{p} \) is the sample proportion, \( p_0 \) is the hypothesized population proportion, and \( n \) is the sample size.

A higher modulus of the z-score indicates a larger deviation of the sample proportion from the hypothesized value, providing evidence against the null hypothesis.
Population Proportion
The population proportion is a parameter that represents the ratio of individuals in a population with a particular characteristic or trait. Hypothesis tests concerning population proportions aim to determine if the observed sample proportion significantly deviates from this known or claimed population proportion. Here's how it functions:
  • The initial step involves specifying the null hypothesis that claims the population proportion equals a certain value, say \( p = 0.25 \).
  • A random sample is drawn, and the sample proportion \( \hat{p} \) is calculated.
  • The test statistic, typically a z-score, is calculated to see how far \( \hat{p} \) diverges from the hypothesized \( p \).

This comparison helps in deciding whether to reject the null hypothesis, depending on the test type and the critical values involved. Population proportion testing is crucial in fields like market research, biology, and quality control.

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Most popular questions from this chapter

Make the following hypothesis tests about \(p\). a. \(H_{0}: p=.57, \quad H_{1}: p \neq .57, \quad n=800, \quad \hat{p}=.50, \quad \alpha=.05\) b. \(H_{0}=p=.26, \quad H_{1}: p<.26, \quad n=400, \quad \hat{p}=.23, \quad \alpha=.01\) c. \(H_{0}=p=.84, \quad H_{1}: p>.84, \quad n=250, \quad \hat{p}=.85, \quad \alpha=.025\)

Perform the following tests of hypothesis. a. \(H_{0}: \mu=285, \quad H_{1}: \mu<285\) \(n=55, \quad \bar{x}=267.80, \quad s=42.90, \quad \alpha=.05\) b. \(H_{0-\mu}=10.70, \quad H_{1}: \mu \neq 10.70, \quad n=47, \bar{x}=12.025, \quad s=4.90, \quad \alpha=.01\) c. \(H_{0}=\mu=147,500, \quad H_{1}: \mu>147,500, n=41, \bar{x}=149,812, s=22,972, \alpha=.10\)

The mean balance of all checking accounts at a bank on December 31,2011, was \(\$ 850 .\) A random sample of 55 checking accounts taken recently from this bank gave a mean balance of \(\$ 780\) with a standard deviation of \(\$ 230 .\) Using a \(1 \%\) significance level, can you conclude that the mean balance of such accounts has decreased during this period? Explain your conclusion in words. What if \(\alpha=.025\) ?

Consider the null hypothesis \(H_{0}=\mu=625 .\) Suppose that a random sample of 29 observations is taken from a normally distributed population with \(\sigma=32 .\) Using a significance level of \(.01\), show the rejection and nonrejection regions on the sampling distribution curve of the sample mean and find the critical value(s) of \(z\) when the alternative hypothesis is as follows. a. \(H_{1}: \mu \neq 625 \quad\) b. \(H_{1}: \mu>625 \quad\) c. \(H_{1}: \mu<625\)

Make the following tests of hypotheses. a. \(H_{0}: \mu=80, \quad H_{1}: \mu \neq 80, \quad n=33, \quad \bar{x}=76.5, \quad \sigma=15, \quad \alpha=.10\) b. \(H_{0}=\mu=32, \quad H_{1}: \mu<32, \quad n=75, \quad \bar{x}=26.5, \quad \sigma=7.4, \quad \alpha=.01\) c. \(H_{0}=\mu=55, \quad H_{1}: \mu>55, \quad n=40, \bar{x}=60.5, \quad \sigma=4, \quad \alpha=.05\)
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