Are you smarter than a second-grader? A random sample of 

58

 second-graders in a certain school district are given a standardized mathematics skills test. The sample mean score is =

x

54

. Assume the standard deviation of test scores is =

σ

13

. The nationwide average score on this test is 58

. The school superintendent wants to know whether the second-graders in her school district have weaker math skills than the nationwide average. Use the =

α

0.01

 level of significance and the critical value method.

Part 1 of 5





(a) State the appropriate null and alternate hypotheses.

:

H

0

 =

μ

58



:

H

1

 <

μ

58



This hypothesis test is a ▼

left-tailed

 test.

Part: 1 / 5

1 of 5 Parts Complete

Part 2 of 5

Find the critical value(s). Round the answer(s) to three decimal places, if necessary. If there is more than one critical value, separate them with commas.

Critical value(s): −








To find the critical value for this hypothesis test, we need to use the following details:

1. **Significance level** \( \alpha = 0.01 \).
2. **Hypotheses**:
   - Null hypothesis: \( H_0: \mu = 58 \)
   - Alternative hypothesis: \( H_1: \mu < 58 \)
3. This is a **left-tailed test** because we are testing if the mean is less than the nationwide average.

For a left-tailed test with \( \alpha = 0.01 \), we will use the **z-distribution** because the population standard deviation \( \sigma = 13 \) is known.

### Step-by-Step Solution

1. **Determine the critical z-value**:
   - For a left-tailed test at a \(0.01\) significance level, we find the z-score that has a cumulative probability of \(0.01\).
   - Looking up the z-score for \(0.01\) in a z-table or using statistical software, we get:

\[
z_{\alpha} = -2.326
\]

### Answer
The critical value for this test is:
\[
-2.326
\]

Would you like to continue with further steps or have any questions?

---

**5 Relative Questions to Consider:**

1. How do we find the test statistic for this hypothesis test?
2. How do we interpret the results if the test statistic falls below the critical value?
3. How does changing the significance level \( \alpha \) affect the critical value?
4. Why is the z-distribution appropriate in this scenario instead of the t-distribution?
5. How would we approach this test if the sample size were much smaller?

**Tip:** Always check if the sample size is large or the population standard deviation is known, as these factors determine whether to use a z-test or t-test.

Are you smarter than a second-grader? A random sample of 58 second-graders in a certain school district are given a standardized mathematics skills test. The sample mean score is 54. Assume the standard deviation of test scores is 13. The nationwide average score on this test is 58. The school superintendent wants to know whether the second-graders in her school district have weaker math skills than the nationwide average. Use the α = 0.01 level of significance and the critical value method.

Analyze whether second-graders in a certain district have weaker math skills compared to the nationwide average using hypothesis testing and the critical value method. With a sample size of 58, a standard deviation of 13, and a significance level of α = 0.01, this problem applies the z-test for a left-tailed hypothesis test.

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