A decade-old study found that the proportion, p, of high school seniors who believed that "getting rich" was an important personal goal was 75%. A researcher decides to test whether or not that percentage still stands. He finds that, among the 245 high school seniors in his random sample, 194 believe that "getting rich" is an important goal. Can he conclude, at the 0.01 level of significance, that the proportion has indeed changed?

Perform a two-tailed test. Then complete the parts below.

Carry your intermediate computations to three or more decimal places. (If necessary, consult a list of formulas.)

(a) State the null hypothesis H_{0} and the alternative hypothesis H_{2}

H_{0}

H_{1}

(b) Determine the type of test statistic to use.

(c) Find the value of the test statistic. (Round to three or more decimal places.)

(d) Find the p-value. (Round to three or more decimal places.)

(e) Can we conclude that the proportion of high school seniors who believe that "getting rich" is an important goal has changed?

Yes

No

To solve this problem, we will perform a hypothesis test for a population proportion. Let's go through the steps.

### (a) State the null hypothesis \( H_0 \) and the alternative hypothesis \( H_1 \)

- **Null Hypothesis (\( H_0 \))**: The proportion of high school seniors who believe that "getting rich" is an important goal is still 75%.
  \[
  H_0: p = 0.75
  \]

- **Alternative Hypothesis (\( H_1 \))**: The proportion of high school seniors who believe that "getting rich" is an important goal has changed from 75%.
  \[
  H_1: p \neq 0.75
  \]
  
This is a two-tailed test because we are testing whether the proportion has changed (could be either higher or lower).

### (b) Determine the type of test statistic to use

Since we are testing a hypothesis about a population proportion, the appropriate test statistic is the **z-test statistic**.

### (c) Find the value of the test statistic

The formula for the z-test statistic for a population proportion is given by:
\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]
Where:
- \(\hat{p}\) is the sample proportion
- \(p_0\) is the hypothesized population proportion
- \(n\) is the sample size

Given:
- \( \hat{p} = \frac{194}{245} \)
- \( p_0 = 0.75 \)
- \( n = 245 \)

First, let's calculate the sample proportion \( \hat{p} \):

\[
\hat{p} = \frac{194}{245} \approx 0.792
\]

Now, we can calculate the z-value:

\[
z = \frac{0.792 - 0.75}{\sqrt{\frac{0.75 \times 0.25}{245}}}
\]

Let's compute this value.

### (d) Find the p-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true.

For a two-tailed test, the p-value is computed as:
\[
\text{p-value} = 2 \times P(Z > |z|)
\]

### (e) Can we conclude that the proportion of high school seniors who believe that "getting rich" is an important goal has changed?

To determine this, we'll compare the p-value with the significance level \( \alpha = 0.01 \). If the p-value is less than 0.01, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Now, I'll calculate the z-value and the p-value.### Calculation Results:

- **Sample proportion** (\( \hat{p} \)): \( 0.792 \) (rounded to three decimal places)
- **Test statistic** (\( z \)): \( 1.512 \) (rounded to three decimal places)
- **p-value**: \( 0.130 \) (rounded to three decimal places)

### (e) Conclusion

Since the p-value (\( 0.130 \)) is greater than the significance level (\( \alpha = 0.01 \)), we **fail to reject** the null hypothesis. 

**Conclusion:** We do not have sufficient evidence at the 0.01 level of significance to conclude that the proportion of high school seniors who believe that "getting rich" is an important goal has changed.

---

Would you like further details or have any other questions? Here are some related questions you might find helpful:

1. What happens if the significance level is increased to 0.05?
2. How would the result change if the sample size were larger?
3. How is the power of the test related to the sample size and significance level?
4. What is the difference between a one-tailed and a two-tailed test?
5. How can we interpret the z-value in the context of this problem?

**Tip:** Always remember to check whether your p-value is for a one-tailed or two-tailed test, as this can significantly impact your conclusion.

Explore a detailed example of hypothesis testing for a population proportion. Learn how to determine if there is sufficient evidence to conclude that the proportion of high school seniors who believe that "getting rich" is an important goal has changed. Includes calculations of sample proportion, test statistic, and p-value.

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