91Ó°ÊÓ

In \(1994,52 \%\) of parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. A recent survey found that 256 of 800 parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. Do parents feel differently today than they did in \(1994 ?\) (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 \(\alpha=0.05\) level of significance, determine the probability of making a Type II error if the true population proportion is 0.50. What is the power of the test? (c) Redo part (b) if the true proportion is 0.48 .

Step by step solution

01

Define Hypotheses

Define the null and alternative hypotheses. Let the population proportion in 1994 be denoted as \(p_0\).- Null hypothesis \(H_0\): \(p = 0.52\)- Alternative hypothesis \(H_1\): \(p eq 0.52\)

02

Explain Type II Error

A Type II error occurs when the null hypothesis \(H_0\) is false, but fails to be rejected. For this problem, a Type II error would mean concluding that the proportion of parents who think it's a serious problem has not changed from 1994 (\(p = 0.52\)), even though it actually has changed.

03

Calculate Test Statistic

First, calculate the sample proportion \(\hat{p}\) from the data. Given 256 out of 800 parents feel it's a serious problem:\[\hat{p} = \frac{256}{800} = 0.32\]Next, calculate the test statistic for the hypothesis test:\[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} = \frac{0.32 - 0.52}{\sqrt{\frac{0.52(1 - 0.52)}{800}}}\]

04

Find Critical Values

For a two-tailed test at the \(\alpha=0.05\) level of significance, the critical z-values are \(\pm 1.96\).

05

Determine Probability of Type II Error \(\beta\)

For (b), when \(p = 0.50\):- The standard error under \(p = 0.50\) is\[\sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.50(1 - 0.50)}{800}} = \sqrt{\frac{0.25}{800}} = 0.0177\]- Calculate the z-values for the acceptance region under \(H_1\):\[z_{lower} = \frac{-1.96 - (-0.02)}{0.0177}\]\[z_{upper} = \frac{1.96 - (-0.02)}{0.0177}\]Find \(\beta\) as the probability corresponding to these z-values from the standard normal distribution.

06

Calculate Power

The power of the test is \(1 - \beta\). Use the z-table to find the probabilities corresponding to the calculated z-values and determine the cumulative probability bounded by these z-values.

07

Redo for \(p = 0.48\)

Repeat Steps 5 and 6 with \(p = 0.48\).- Standard error under \(p = 0.48\) is\[\sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.48(1 - 0.52)}{800}} = 0.0176\]- Find corresponding z-values and then calculate power.

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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 happens when we fail to reject a false null hypothesis. In simpler words, it's like saying 'nothing has changed' when in fact something has. In our example, a Type II error would be concluding that the proportion of parents who think it's a serious problem hasn't changed from 1994, even though it actually has. It’s crucial because if we make this error, we might miss important changes in public opinion. To minimize this risk, we need to ensure our tests are sensitive enough to detect real changes.

Population proportion

Population proportion represents the fraction of individuals in a population that have a particular characteristic. For example, if 52% of parents in 1994 felt the lack of math and science education was serious, this percentage (52%) is the population proportion. In our current survey, we sample 800 parents, finding that 256 out of 800, or 32%, share this concern.
Comparing these sample proportions helps us understand if today's parents feel differently than those in 1994.

Test Statistic Calculation

To determine if there’s a significant change in proportions, we calculate a test statistic. First, we find the sample proportion: \(\frac{256}{800} = 0.32\).
Next, we use the formula for the test statistic:
\[ z = \frac{\bar{p} - p_0}{\frac{\text{standard error}}{n}} \]
Here, \( \bar{p} \) is our sample proportion (0.32), \( p_0 \) is the population proportion (0.52), and the standard error is computed as:
\[ \text{SE} = \frac{\text{standard deviation}}{\text{square root of n}}\]
This gives us the z-score, which tells us how many standard deviations away our sample proportion is from the population proportion.

Critical values

Critical values are threshold points that help determine whether to reject the null hypothesis. For a two-tailed test at the \( \alpha = 0.05 \) significance level, the critical z-values are \( \pm 1.96 \).
If our calculated test statistic falls beyond these critical values, we reject the null hypothesis, suggesting a significant change in opinion since 1994.

Standard Error

Standard error measures the variability of the sample proportion from the population proportion. It's calculated as:
\[ \text{SE} = \frac{\text{Standard deviation}}{\text{square root of n}}\]
In our case, for a given population proportion \( p \) (e.g., 0.50 or 0.48), the standard error helps us understand the spread of sample proportions around the true population proportion:
\[ \text{SE} = \frac{\text{sqrt}(p \times (1 - p) / n)}\]
This calculation informs our test decisions by creating a margin of error around our sample proportion.