To test Upper H 0 : sigma equals 53 versus Upper H 1 : sigma less than 53​, a random sample of size n equals 21 is obtained from a population that is known to be normally distributed. Complete parts​ (a) through​ (c).
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Part 1
​(a) If the sample standard deviation is determined to be s equals 50.6​, compute the test statistic.
chi Subscript 0 Superscript 2equals
  
enter your response here ​(Round to two decimal places as​ needed.)

The test statistic for this hypothesis test is calculated using the chi-square formula for variance:

\[
\chi^2 = \frac{(n - 1) \cdot s^2}{\sigma^2}
\]

Where:
- \(n = 21\) (sample size)
- \(s = 50.6\) (sample standard deviation)
- \(\sigma = 53\) (population standard deviation under \(H_0\))

### Step 1: Calculate \(s^2\) and \(\sigma^2\):
\[
s^2 = 50.6^2 = 2560.36
\]
\[
\sigma^2 = 53^2 = 2809
\]

### Step 2: Plug values into the formula for the test statistic:
\[
\chi^2 = \frac{(21 - 1) \cdot 2560.36}{2809}
\]
\[
\chi^2 = \frac{20 \cdot 2560.36}{2809} = \frac{51207.2}{2809}
\]
\[
\chi^2 \approx 18.23
\]

So, the test statistic is approximately **18.23**.

Would you like to see the next steps or have any questions?

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Here are 5 related questions for practice:

1. How do you find the critical value for a chi-square test?
2. How would the test statistic change if the sample standard deviation were higher than 53?
3. What is the degree of freedom in this test?
4. How do you interpret the result of a chi-square test?
5. Can a chi-square test be used for non-normally distributed data?

**Tip:** The degrees of freedom in chi-square tests for variance are always \(n-1\).

To test Upper H 0 : sigma equals 53 versus Upper H 1 : sigma less than 53​, a random sample of size n equals 21 is obtained from a population that is known to be normally distributed. If the sample standard deviation is determined to be s equals 50.6​, compute the test statistic.

Learn how to compute the test statistic for a chi-square test with a sample standard deviation of 50.6 and a population standard deviation of 53. This example covers key statistical concepts such as variance calculation, hypothesis testing, and chi-square distribution. Suitable for undergraduate statistics courses.

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