Find the value of the standard​ score, z, and determine whether to reject the null hypothesis at a 0.10 significance level. Is the alternative hypothesis​ supported?
Upper H 0​: mu equals 20.0 ​seconds, Upper H Subscript a​: mu greater than 20.0 ​seconds, n equals 144​, x overbar equals 20.7 ​seconds, sigma equals 3.5 seconds

To solve this problem, we need to calculate the standard score (z-score) and determine whether to reject the null hypothesis at a 0.10 significance level.

### Step 1: State the hypotheses
- **Null Hypothesis (H₀):** \( \mu = 20.0 \) seconds
- **Alternative Hypothesis (Hₐ):** \( \mu > 20.0 \) seconds

### Step 2: Given data
- Population mean (\( \mu_0 \)): 20.0 seconds
- Sample mean (\( \bar{x} \)): 20.7 seconds
- Population standard deviation (\( \sigma \)): 3.5 seconds
- Sample size (\( n \)): 144
- Significance level (\( \alpha \)): 0.10

### Step 3: Calculate the standard score (z-score)
The z-score is calculated using the formula:

\[
z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]

Plugging in the values:

\[
z = \frac{20.7 - 20.0}{\frac{3.5}{\sqrt{144}}}
\]

\[
z = \frac{0.7}{\frac{3.5}{12}} = \frac{0.7}{0.2917} \approx 2.4
\]

### Step 4: Determine the critical z-value and make a decision
Since the alternative hypothesis is \( \mu > 20.0 \), this is a one-tailed test. We look for the critical z-value at the 0.10 significance level:

- The critical z-value for a 0.10 significance level in a one-tailed test is approximately 1.28.

### Step 5: Compare the calculated z-value to the critical z-value
- Calculated z-value: 2.4
- Critical z-value: 1.28

Since 2.4 > 1.28, we reject the null hypothesis.

### Conclusion:
- **Reject the null hypothesis** at the 0.10 significance level.
- The alternative hypothesis \( \mu > 20.0 \) seconds **is supported**.

Would you like more details or have any questions?

Here are some related questions:

1. How does the z-score change if the sample mean increases?
2. What would be the result if the significance level was set at 0.05?
3. How would the result change if the sample size was smaller, say 100?
4. What is the probability of Type I error in this hypothesis test?
5. How do we interpret a z-score of 2.4 in this context?

**Tip:** In hypothesis testing, the critical value approach is common, but the p-value approach is also widely used, giving a more precise measure of the strength of evidence against the null hypothesis.

Learn how to calculate the z-score and perform a hypothesis test to determine if the population mean is greater than 20.0 seconds. This problem involves using the z-score formula with given population mean, sample mean, population standard deviation, and sample size. Suitable for college-level students and statistical analysis enthusiasts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation