Questions: Question 7 The probability of rejecting the null hypothesis when, in fact, the null hypothesis is true is called the standard error p-value power of the test significance level Question 8 What is the statistic that compares the observed proportion to the expected proportion? the one-proportion t-test statistic the one-proportion z-test statistic the standard error

Question 7
The probability of rejecting the null hypothesis when, in fact, the null hypothesis is true is called the
standard error
p-value
power of the test
significance level

Question 8
What is the statistic that compares the observed proportion to the expected proportion?
the one-proportion t-test statistic
the one-proportion z-test statistic
the standard error

Transcript text: Question 7 The probability of rejecting the null hypothesis when, in fact, the null hypothesis is true is called the standard error $p$-value power of the test significance level Question 8 What is the statistic that compares the observed proportion to the expected proportion? the one-proportion t-test statistic the one-proportion z-test statistic the standard error

Solution

Solution Steps

Step 1: Hypothesis Test Setup

We are conducting a hypothesis test for a population proportion. The null hypothesis $ H_0 $ states that the population proportion $ p $ is equal to the hypothesized value $ p_0 = 0.5 $. The alternative hypothesis $ H_a $ states that the population proportion is not equal to $ p_0 $.

Step 2: Test Statistic Calculation

The test statistic $ Z $ is calculated using the formula:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

where:

$ \hat{p} = 0.55 $ (sample proportion),
$ p_0 = 0.5 $ (hypothesized population proportion),
$ n = 100 $ (sample size).

Substituting the values, we find:

\[ Z = \frac{0.55 - 0.5}{\sqrt{\frac{0.5(1 - 0.5)}{100}}} = 1.0 \]

Step 3: P-value Calculation

The P-value associated with the test statistic $ Z = 1.0 $ is calculated to be $ 0.3173 $. This value indicates the probability of observing a sample proportion as extreme as $ 0.55 $ under the null hypothesis.

Step 4: Critical Region Determination

For a two-tailed test with a significance level $ \alpha = 0.05 $, the critical regions are defined as:

\[ Z < -1.96 \quad \text{or} \quad Z > 1.96 \]

Step 5: Conclusion

Since the calculated test statistic $ Z = 1.0 $ does not fall into the critical region, we fail to reject the null hypothesis. The P-value $ 0.3173 $ is greater than $ \alpha = 0.05 $, further supporting this conclusion.