For a test of 

Upper H 0H0​: 

pequals=​0.50, 

the sample proportion is 

0.470.47 

based on a sample size of 100. Use this information to complete parts ​(a) through ​(c) below.

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Part 1

**a.** Find the test statistic z.

zequals=enter your response here

To find the test statistic \( z \) for the hypothesis test, you can use the formula:

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

where:
- \( \hat{p} \) is the sample proportion
- \( p_0 \) is the hypothesized population proportion (from \( H_0: p = 0.50 \))
- \( n \) is the sample size

Given:
- \( \hat{p} = 0.47 \)
- \( p_0 = 0.50 \)
- \( n = 100 \)

Let's calculate the test statistic \( z \):

\[
z = \frac{0.47 - 0.50}{\sqrt{\frac{0.50(1 - 0.50)}{100}}}
\]

Now, we can compute this step-by-step:

1. Calculate the numerator: \( 0.47 - 0.50 = -0.03 \).
2. Calculate the denominator: 

\[
\sqrt{\frac{0.50(1 - 0.50)}{100}} = \sqrt{\frac{0.50 \times 0.50}{100}} = \sqrt{\frac{0.25}{100}} = \sqrt{0.0025} = 0.05
\]

3. Now, divide the numerator by the denominator:

\[
z = \frac{-0.03}{0.05} = -0.60
\]

So, the test statistic \( z \) is \(-0.60\).

Would you like more details or have any further questions?

Here are 8 related questions you might find helpful:
1. How do you interpret the \( z \)-value in a hypothesis test?
2. What is the next step after finding the test statistic in hypothesis testing?
3. How do you determine the p-value from the test statistic \( z \)?
4. What is the critical value for a two-tailed test at a 5% significance level?
5. How does the sample size affect the \( z \)-value in hypothesis testing?
6. What are the assumptions for using a \( z \)-test for proportions?
7. How would the test statistic change if the sample proportion were 0.53 instead of 0.47?
8. What are the differences between a \( z \)-test and a \( t \)-test? 

**Tip:** When the sample size is large (generally \( n > 30 \)), the sampling distribution of the sample proportion can be approximated by a normal distribution, allowing the use of the \( z \)-test.

Learn how to calculate the test statistic z for hypothesis testing when given a sample proportion and a hypothesized population proportion. This problem involves a sample proportion of 0.47 from a sample size of 100, testing against a hypothesized proportion of 0.50. Follow step-by-step calculations to understand the process.

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