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Chapter 10: Problem 3

In Problems \(3-8,\) test the hypothesis, using (a) the classical approach and then (b) the P-value approach. Be sure to verify the requirements of the test. $$\begin{aligned}&H_{0}: p=0.3 \text { versus } H_{1}: p>0.3\\\&n=200 ; x=75 ; \alpha=0.05\end{aligned}$$

Short Answer

Expert verified
Reject the null hypothesis. There is sufficient evidence that the population proportion \( p \) is greater than 0.3.

Step by step solution

01

Define the null and alternative hypotheses

The problem gives us the null hypothesis as \( H_0: p = 0.3 \) and the alternative hypothesis as \( H_1: p > 0.3 \).
02

Check the test requirements

For hypothesis testing of a proportion, the sample size \( n \) and expected number of successes and failures should be large enough. Ensure that both \( np \) and \( n(1 - p) \) are greater than 5. Here, \( np = 200 \times 0.3 = 60 \) and \( n(1 - p) = 200 \times 0.7 = 140 \), both greater than 5.
03

Calculate the test statistic (Classical Approach)

Use the test statistic for a proportion: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] Here, \( \hat{p} \) is the sample proportion: \( \hat{p} = \frac{x}{n} = \frac{75}{200} = 0.375 \). Substituting the values: \[ z = \frac{0.375 - 0.3}{\sqrt{\frac{0.3(1 - 0.3)}{200}}} = \frac{0.075}{0.0324} \approx 2.31 \].
04

Determine the critical value and compare (Classical Approach)

For \( \alpha = 0.05 \), the critical value for a one-tailed test is \( z_{\alpha} = 1.645 \). Since the calculated test statistic \( z = 2.31 \) is greater than \( 1.645 \), reject the null hypothesis.
05

Calculate the P-value (P-Value Approach)

The P-value is the probability of obtaining a test statistic at least as extreme as the observed result, assuming that the null hypothesis is true. Using the standard normal distribution table, a \( z \) value of 2.31 gives a P-value of approximately 0.0104.
06

Compare the P-value to \( \alpha \) (P-Value Approach)

Since the P-value (0.0104) is less than \( \alpha = 0.05 \), we reject the null hypothesis.
07

State the conclusion

Based on both the classical and P-value approaches, we reject the null hypothesis. There is sufficient evidence to support the claim that the population proportion \( p \) is greater than 0.3.

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Key Concepts

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

Null Hypothesis
When you start a hypothesis test, you first need to set up your hypotheses. The null hypothesis (denoted as \(H_0\)) is a statement of no effect or no difference. It represents the idea that any observed difference is due to random chance. In our exercise, the null hypothesis is \(H_0: p = 0.3\). This means we are assuming the proportion of success (p) in the population is 0.3 until we have enough evidence to prove otherwise.
Alternative Hypothesis
The alternative hypothesis (denoted as \(H_1\)) directly opposes the null hypothesis. It represents the effect or difference we are testing for. For our example, the alternative hypothesis is \(H_1: p > 0.3\). This means we believe the proportion of success is greater than 0.3. If we gather enough evidence, we'll reject the null hypothesis in favor of this alternative.
Test Statistic
To test our hypotheses, we calculate a test statistic, which measures how far our sample result is from the null hypothesis. For proportions, we use a z-test statistic: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] In our exercise, the sample proportion \( \hat{p} \) is 0.375 and the hypothesized proportion \( p_0 \) is 0.3. Substituting into our formula, we get \[ z = \frac{0.375 - 0.3}{\sqrt{\frac{0.3(1 - 0.3)}{200}}} = \frac{0.075}{0.0324} \approx 2.31 \]A higher test statistic means it's less likely the null hypothesis is true.
P-value
The P-value measures the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed if the null hypothesis is true. For a test statistic of 2.31, the P-value is approximately 0.0104. This P-value tells us there is a 1.04% chance that the results we obtained could happen under the null hypothesis. If the P-value is less than the significance level (伪), which is 0.05 in this case, we reject the null hypothesis. Since 0.0104 < 0.05, the evidence is strong enough to reject \(H_0\).
Critical Value
The critical value is the threshold at which we reject the null hypothesis. For our one-tailed test with \( \alpha = 0.05 \), the critical value is \( z_{\alpha} = 1.645 \). This value means that if our test statistic is greater than 1.645, we'll reject the null hypothesis. In our case, the calculated test statistic (2.31) is greater than 1.645, which means we reject the null hypothesis. This is another way to see that our sample provides significant evidence against \(H_0\).

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