91Ó°ÊÓ

In August \(2002,47 \%\) of parents who had children in grades \(\mathrm{K}-12\) were satisfied with the quality of education the students receive. A recent Gallup poll found that 437 of 1013 parents of children in grades \(\mathrm{K}-12\) were satisfied with the quality of education the students receive. Does this suggest the proportion of parents satisfied with the quality of education has decreased? (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the \(\alpha=0.10\) level of significance, determine the probability of making a Type II error if the true population proportion is 0.42. What is the power of the test? (c) Redo part (b) if the true proportion is 0.46 .

Step by step solution

01

Title - Define Hypotheses

We need to set up our null and alternative hypotheses. Let's define \[H_0: p = 0.47 \] (The proportion of satisfied parents is still 47%.) \[H_1: p < 0.47 \] (The proportion of satisfied parents has decreased.)

02

Title - Calculate Sample Proportion

The sample proportion \( \hat{p} \) is calculated by dividing the number of satisfied parents by the total number of parents surveyed: \[ \hat{p} = \frac{437}{1013} = 0.431 \].

03

Title - Interpret Type II Error (Part a)

A Type II error occurs when we fail to reject the null hypothesis (we think the proportion has not decreased) when in fact it is false (the proportion has indeed decreased).

04

Title - Determine Standard Error

Calculate the standard error (SE) using the hypothesized population proportion: \[ SE = \sqrt{ \frac{0.47 (1 - 0.47)}{1013} } = 0.0157 \].

05

Title - Find Z-Score for Critical Value

For \( \alpha = 0.10 \), the critical Z-value (\( Z_{\alpha} \)) is found using a Z-table or calculator: \[ Z_{\alpha} = -1.28 \] (one-tailed test).

06

Title - Calculate Critical Proportion

The critical value for the sample proportion (p-critical) is computed as: \[ p_{\text{critical}} = 0.47 + (-1.28) \times 0.0157 = 0.4499 \].

07

Title - Compute Z for Type II Error (Part b with p = 0.42)

Calculate the Z-score at \( p = 0.42 \) using the standard error: \[ Z = \frac{0.4499 - 0.42}{0.0157} = 1.92 \].Using the Z-table, the probability of Z being less than 1.92 is 0.9726.

08

Title - Type II Error Probability and Power (Part b)

Thus, the probability of making a Type II error \( \beta = 1 - 0.9726 = 0.0274 \). The power of the test is \( 1 - \beta = 0.9726 \).

09

Title - Redo with p = 0.46 for Part c

For \( p = 0.46 \): Calculate Z-score: \[ Z = \frac{0.4499 - 0.46}{0.0157} = -0.64 \].From the Z-table, the probability of Z being less than -0.64 is 0.2611.

10

Title - Type II Error Probability and Power (Part c)

Thus, \( \beta = 1 - 0.2611 = 0.7389 \) and the power of the test is \( 1 - \beta = 0.2611 \).

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Key Concepts

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

Type II error

A Type II error occurs when we fail to reject the null hypothesis even though it is false. In simpler terms, we mistakenly conclude that there is no effect or difference when there actually is one. For example, in the exercise, if the true proportion of parents satisfied with the quality of education is less than 47% but our test fails to show this, we are committing a Type II error. This can be problematic in decision-making processes because we might miss out on identifying and addressing significant issues.

standard error

The standard error (SE) measures the accuracy with which a sample represents a population. It is calculated using the formula: \[ SE = \sqrt{\frac{p (1 - p)}{n}} \] where \( p \) is the hypothesized population proportion and \( n \) is the sample size. In our exercise, using \( p = 0.47 \) and \( n = 1013 \), we get: \[ SE = \sqrt{\frac{0.47 * (1 - 0.47)}{1013}} \approx 0.0157 \]. This value tells us how much variability we can expect in our sample proportion from the true population proportion. A smaller standard error indicates a more precise estimate from our sample.

sample proportion

The sample proportion is the fraction of individuals in the sample with a specific characteristic. For our problem, the sample proportion (\( \hat{p} \)) is calculated as: \[ \hat{p} = \frac{437}{1013} \approx 0.431 \]. This means roughly 43.1% of the surveyed parents are satisfied with the quality of their children's education. Comparing the sample proportion to the hypothesized population proportion (47%) can help determine if there has been a significant change in satisfaction levels.

statistical power

Statistical power is the probability of correctly rejecting the null hypothesis when it is false. High power means there is a higher chance of detecting a real effect. The power of a test is calculated as \( 1 - \beta \), where \( \beta \) is the probability of making a Type II error. In part (b) of our problem, with a true proportion of 0.42, we found: \[ \beta = 0.0274 \] and hence, the power: \[ 1 - \beta = 0.9726 \]. For part (c), with a true proportion of 0.46, we found \( \beta = 0.7389 \) and the power: \[ 1 - \beta = 0.2611 \]. Higher power values (closer to 1) are preferable as they indicate greater sensitivity to detect true effects.

significance level

The significance level (alpha, \( \alpha \)) is the threshold used to decide whether to reject the null hypothesis. It represents the probability of committing a Type I error, which is rejecting the null hypothesis when it is actually true. In our exercise, the researcher decides to use \( \alpha = 0.10 \). This means there is a 10% risk of falsely concluding that the proportion of satisfied parents has decreased. The choice of \( \alpha \) impacts the critical value for the test. For example, with \( \alpha = 0.10 \), the critical Z-value for one-tailed test is -1.28, setting the boundary for rejecting the null hypothesis.