Examine the following hypothesis test with nequals=1818​, x overbarxequals=70.970.9​, sequals=7.87.8​, and alphaαequals=0.010.01.
H Subscript 0​: muμ
less than or equals≤
7070
H Subscript A​: muμ
greater than>
7070
a. State the decision rule in terms of the critical value of the test statistic.
b. State the calculated value of the test statistic.
c. State the conclusion.
Question content area bottom
Part 1
a. State the decision rule. Select the correct choice below and fill in any answer boxes in your choice.
​(Round to four decimal places as​ needed.)
A.
Reject the null hypothesis if the calculated value of the test​ statistic, t, is greater than the critical value of enter your response here. ​Otherwise, do not reject.
B.
Reject the null hypothesis if the calculated value of the test​ statistic, t, is less than the critical value of enter your response here. ​Otherwise, do not reject.
C.
Reject the null hypothesis if the calculated value of the test​ statistic, t, is less than the critical value of enter your response here or greater than the critical value of enter your response here. ​Otherwise, do not reject.

Let's break down the hypothesis test and solve the different parts step by step:

### Given Information:
- Sample size \( n = 1818 \)
- Sample mean \( \overline{x} = 70.9 \)
- Sample standard deviation \( s = 7.8 \)
- Significance level \( \alpha = 0.01 \)
- Null hypothesis \( H_0: \mu \leq 70 \)
- Alternative hypothesis \( H_A: \mu > 70 \)

This is a **one-tailed t-test** because the alternative hypothesis suggests that \( \mu \) is greater than 70.

#### Part a. State the decision rule.
The decision rule is based on the **critical value** from the t-distribution, which depends on the degrees of freedom and the significance level (\( \alpha \)).

- The degrees of freedom \( df = n - 1 = 1818 - 1 = 1817 \).
- The critical value \( t_{\alpha, df} \) is determined from a t-table or statistical software. For a right-tailed test at \( \alpha = 0.01 \), we look for the critical value for \( df = 1817 \). Since degrees of freedom are quite large, we can approximate it using the standard normal distribution \( Z \), but for precision, we use the t-distribution table.

So the decision rule will be:
- **A.** Reject the null hypothesis if the calculated value of the test statistic \( t \) is greater than the critical value. Otherwise, do not reject.

#### Part b. State the calculated value of the test statistic.

The test statistic for a t-test is calculated using the following formula:

\[
t = \frac{\overline{x} - \mu_0}{\frac{s}{\sqrt{n}}}
\]

Where:
- \( \overline{x} = 70.9 \) (sample mean),
- \( \mu_0 = 70 \) (hypothesized population mean under \( H_0 \)),
- \( s = 7.8 \) (sample standard deviation),
- \( n = 1818 \) (sample size).

Let's calculate the value of the t-statistic:

\[
t = \frac{70.9 - 70}{\frac{7.8}{\sqrt{1818}}}
\]

\[
t = \frac{0.9}{\frac{7.8}{42.643}} = \frac{0.9}{0.183} \approx 4.9180
\]

So the calculated value of the test statistic \( t \approx 4.9180 \).

#### Part c. State the conclusion.

Now, we compare the calculated test statistic \( t = 4.9180 \) with the critical value at \( \alpha = 0.01 \).

- From a t-table (or statistical software), for a one-tailed test with \( df = 1817 \) at \( \alpha = 0.01 \), the critical value \( t_{\alpha, df} \) is approximately **2.326** (since it's close to the standard normal critical value for large \( df \)).

Since \( t = 4.9180 \) is greater than the critical value \( 2.326 \), we **reject the null hypothesis**.

### Conclusion:
We have enough evidence to reject the null hypothesis at the \( \alpha = 0.01 \) level. Therefore, we conclude that the population mean \( \mu \) is significantly greater than 70.

### Would you like further details or have any questions?

### 5 Related Questions:
1. How is the t-distribution different from the normal distribution in hypothesis testing?
2. What happens if the significance level \( \alpha \) is changed to 0.05 instead of 0.01?
3. How do we determine whether to use a one-tailed or two-tailed test?
4. How would the decision change if the sample size were smaller, say \( n = 30 \)?
5. What are Type I and Type II errors in hypothesis testing?

### Tip:
When sample sizes are large (typically \( n > 30 \)), the t-distribution approaches the normal distribution, allowing you to approximate critical values using the standard normal table for faster calculations.

Examine the following hypothesis test with n = 1818, x̄ = 70.9, s = 7.8, and α = 0.01.
H₀: μ ≤ 70
Hₐ: μ > 70
a. State the decision rule in terms of the critical value of the test statistic.
b. State the calculated value of the test statistic.
c. State the conclusion.

Learn how to conduct a one-tailed t-test for the mean with a sample size of 1818, sample mean of 70.9, and standard deviation of 7.8 at a 0.01 significance level. This detailed solution covers the calculation of the test statistic, the decision rule, and the final conclusion. Suitable for college-level statistics students.

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