A magazine claims that 25% of its readers are college students. Of a random sample of 200 readers, 42 are college students. Use a .10 level of significance to test Ho: p =.25 and H1: p ≠ .25 Show the hypothesis test procedure.

To test the hypothesis, we can use the two-sample proportion test. Here's the step-by-step procedure:

Step 1: State the null and alternative hypotheses:
- Null hypothesis (Ho): The proportion of college students in the population of readers is equal to 25% (p = 0.25).
- Alternative hypothesis (H1): The proportion of college students in the population of readers is not equal to 25% (p ≠ 0.25).

Step 2: Set the significance level:
- The given significance level is 0.10.

Step 3: Collect and organize the data:
- Random sample of 200 readers.
- Out of those, 42 are college students.

Step 4: Calculate the test statistic:
- We need to calculate the Z-score using the formula:
Z = (p̂ - p) / √[(p * (1 - p)) / n]
where p̂ is the sample proportion, p is the hypothesized proportion, and n is the sample size.

In this case, p̂ = 42/200 = 0.21, p = 0.25, and n = 200. Therefore, the Z-score can be computed as follows:
Z = (0.21 - 0.25) / √[(0.25 * (1 - 0.25)) / 200]

Step 5: Determine the critical value:
- Since we are using a two-tailed test, we need to split the significance level (α = 0.10) in half to get two critical values.
- The critical value for α/2 = 0.10/2 = 0.05 can be found using a standard normal distribution table or a statistical calculator.
- Let's assume the critical value is -1.96 and +1.96 for a 95% confidence level.

Step 6: Make a decision:
- If the Z-score calculated in Step 4 falls within the rejection region defined by the critical values, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Step 7: Calculate the p-value (optional):
- If you prefer, you can also calculate the p-value using the Z-score obtained in Step 4. The p-value represents the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true.

Step 8: Draw a conclusion:
- Based on the decision made in Step 6, we can draw a conclusion about the statement of the null and alternative hypotheses at the given significance level.

Please note that the critical values and the calculated Z-score might differ depending on the specifics of the problem.

To test the hypothesis Ho: p = .25 and H1: p ≠ .25, where p represents the proportion of college students among the magazine readers, we can perform a hypothesis test using the following steps:

Step 1: State the null and alternative hypotheses:
- Null Hypothesis (Ho): The proportion of college students among the magazine readers is equal to 0.25 (p = 0.25).
- Alternative Hypothesis (H1): The proportion of college students among the magazine readers is not equal to 0.25 (p ≠ 0.25).

Step 2: Determine the level of significance (α):
The level of significance, denoted by α, is the probability of rejecting the null hypothesis when it is true. In this case, the α is given as 0.10, which means that we are willing to accept a 10% chance of making a Type I error (rejecting the null hypothesis when it is true).

Step 3: Collect the data:
In this case, we have a random sample of 200 readers, and 42 of them are college students.

Step 4: Compute the test statistic:
To determine whether the proportion of college students is significantly different from 0.25, we can use the formula for the test statistic:
Test Statistic (z) = (p̂ - p) / √(p(1-p) / n)
Where:
- p̂ is the sample proportion of college students (in this case, 42/200 = 0.21),
- p is the hypothesized proportion (0.25),
- n is the sample size (200).

Substituting the values, we get:
z = (0.21 - 0.25) / √(0.25(1-0.25) / 200)

Step 5: Determine the critical value(s):
Since we are performing a two-tailed hypothesis test (H1: p ≠ 0.25), we need to check the critical values for a significance level of α/2 on both ends of the distribution. Looking up the critical value from a standard normal distribution table or using statistical software for a 0.10/2 = 0.05 significance level, we find the critical values to be approximately z = ±1.645.

Step 6: Make a decision:
If the absolute value of the computed test statistic (|z|) is greater than the critical value(s) from Step 5, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Step 7: Calculate the p-value:
The p-value is the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true. For a two-tailed test, we calculate the p-value by doubling the area in the tails beyond the critical values.

Step 8: Interpret the results:
If the p-value is less than the significance level (α), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Note: The actual calculations for the test statistic, critical values, and p-value can be determined using statistical software or online calculators.

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