Consider Example 9.6 with the null hypothesis H0: P=P0=0.50...

Consider Example $9.6$ with the null hypothesis $$ H_{0}: P=P_{0}=0.50 $$ and the alternative hypothesis $$ H_{0}: P \neq 0.50 $$ The decision rule is $$ \begin{gathered} \frac{\hat{p}_{x}-0.50}{\sqrt{0.50(1-0.50) / 600}}<-1.96 \text { or } \\ \frac{\hat{p}_{x}-0.50}{\sqrt{0.50(1-0.50) / 600}}>1.96 \end{gathered} $$ with a sample size of $n=600$. What is the probability of Type II error if the actual population proportion is each of the following? a. $P=0.52$ b. $P=0.58$ c. $P=0.53$ d. $P=0.48$. e. $P=0.43$

Consider Example $9.6$ with the null hypothesis
$$
H_{0}: P=P_{0}=0.50
$$
and the alternative hypothesis
$$
H_{0}: P \neq 0.50
$$
The decision rule is
$$
\begin{gathered}
\frac{\hat{p}_{x}-0.50}{\sqrt{0.50(1-0.50) / 600}}<-1.96 \text { or } \\
\frac{\hat{p}_{x}-0.50}{\sqrt{0.50(1-0.50) / 600}}>1.96
\end{gathered}
$$
with a sample size of $n=600$. What is the probability of Type II error if the actual population proportion is each of the following?
a. $P=0.52$
b. $P=0.58$
c. $P=0.53$
d. $P=0.48$.
e. $P=0.43$

Key Concepts

Null and Alternative Hypotheses

This is the foundational framework for hypothesis testing where the null hypothesis (H?) represents a statement of no effect or a baseline assumption (e.g., that the population proportion equals a specific value) and the alternative hypothesis (H?) represents a claim that contradicts the null. The structure sets the stage for determining whether observed sample data is compatible with the null hypothesis or provides sufficient evidence to support the alternative hypothesis.

Critical Region and Decision Rule

The decision rule or critical region outlines the criteria for rejecting the null hypothesis. It is defined based on a predetermined significance level (often using quantiles of a relevant distribution, such as the z-distribution). When the test statistic falls into these regions (e.g., below -1.96 or above 1.96 for a two-tailed test at the 5% significance level), the null hypothesis is rejected.

Normal Approximation of Proportions

When dealing with sample proportions, especially with a sufficiently large sample size, the sampling distribution of the sample proportion can be approximated by a normal distribution due to the Central Limit Theorem. This approximation enables the use of z-tests to compare the sample proportion to the hypothesized population proportion under the null hypothesis.

Power of the Test

The power of a statistical test is the probability that the test correctly rejects the null hypothesis when a specific alternative is true. It is calculated as one minus the probability of a Type II error. Evaluating the power at different actual population proportions demonstrates the test’s sensitivity and its ability to detect deviations from the null hypothesis.

Type II Error

A Type II error occurs when the test fails to reject the null hypothesis even though the alternative hypothesis is true. It represents the probability of not detecting an effect that is actually present. Calculating the probability of a Type II error involves determining the likelihood that the test statistic falls in the acceptance region when the true parameter value differs from the value specified in the null hypothesis.

Transcript

00:01 The probability of type 2 error.

00:05 So for any given population proportion pre -p1 included in alternative hypothesis, the probability beta is making a type 2 error.

00:19 So in decision rule, range of values of sample proportion fades to reject the null hypothesis.

00:29 Here, the null hypothesis is p is equal to p.

00:33 Is equal to 0 .50 and the alternative hypothesis is p is not equal to 0 .50.

00:42 The decision criteria for this problem will be reject null hypothesis if p cap minus p .0 upon root over p .0 multiplied by 1 minus p .0 upon n is less than minus x alpha upon 2 or p cap minus p not upon root over p .0 1 minus p not upon n is greater than or greater than z alpha upon 2 so we are given the null hypothesis is point p is equal to 0 .5 so we have p cap minus 0 .50 upon root over 0 .50 multiplied by 0 .50 upon n is 600 is greater than 1 .96 or if p cap is greater than 0 .50 plus 1 .96 multiplied by root over 0 .5 multiplied by 0 .5 multiplied by 0 .5 0 .5 0 upon 600.

02:40 That is equal to 0 .50 plus 0 .04 that is equal to 0 .54 or p cap is less than 0 .50 minus 0 .04 which is 0 .46.

03:04 So now let us find the probability of type 2 error if the actual proportion p1 given to is 0 .52 using the formula this is formula 1 and this is our formula 2 so let's using the formula 1 that is p is equal to probability of p cap greater than 0 .6 or less than or equal to 0 .54 that is equal to probability to probability of 0 .46 minus p1 upon root over p1 multiplied by 1 minus p1 upon n is greater is less than or equal to z and z is less than or equal to 0 .54 minus sorry we are given the values of p1 and we have n so p1 is 0 .52 upon n is 0 .52 upon n is 600 minus 0 .52 upon root over 0 .52 multiplied by 1 minus 0 .52 upon 600 that is equal to probability of minus 2 .94 is less than or equal to z and z is less than or equal to 0 .98 so the value for so so, beta is the value for this is 0 .8365 minus 0 .0016, that is equal to 0 .8349.

05:45 So the probability of a type 2 error if the actual proportion, population proportion is 0 .8349...

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