In conducting the hypothesis test below, your sample consists of 23 observations, with a mean of 57 and standard deviation of 3.44.

H
0
:
μ
≤
55.1

H
1
:
μ
>
55.1

a) This is a
right-tailed test.
Correct

b) Calculate the test statistic, rounded to 3 decimal places.
t
=

3.162
Incorrect

c) At
α
=
0.1
, the rejection region is

t
<
-1.717
or
t
>
1.717

t
<
-1.321

t
>
1.321

none of the above

Correct

d) The decision is to
fail to reject
H
0
since the test statistic does not fall in the rejection region.
reject
H
0
since the test statistic falls in the rejection region.
fail to reject
H
1
since the test statistic does not fall in the rejection region.
reject
H
1
since the test statistic falls in the rejection region.
none of the above

To solve this hypothesis test, we need to follow these steps:

### Given Data:
- Sample size (\(n\)) = 23
- Sample mean (\(\bar{x}\)) = 57
- Sample standard deviation (\(s\)) = 3.44
- Null hypothesis (\(H_0\)): \(\mu \leq 55.1\)
- Alternative hypothesis (\(H_1\)): \(\mu > 55.1\)
- Significance level (\(\alpha\)) = 0.1

### a) This is a right-tailed test.
- **Correct**: Since \(H_1\) is \(\mu > 55.1\), it is indeed a right-tailed test.

### b) Calculate the test statistic:
The test statistic for the t-test is calculated using the formula:
\[
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
\]
Where:
- \(\bar{x}\) = sample mean = 57
- \(\mu_0\) = population mean under the null hypothesis = 55.1
- \(s\) = sample standard deviation = 3.44
- \(n\) = sample size = 23

Substitute the values:
\[
t = \frac{57 - 55.1}{3.44 / \sqrt{23}}
\]

First, calculate the standard error (SE):
\[
SE = \frac{3.44}{\sqrt{23}} \approx \frac{3.44}{4.795} \approx 0.718
\]

Now, calculate the t-value:
\[
t = \frac{57 - 55.1}{0.718} \approx \frac{1.9}{0.718} \approx 2.646
\]

Therefore, the correct test statistic rounded to three decimal places is:
\[
t \approx 2.646
\]

### c) At \(\alpha = 0.1\), the rejection region:
- We need to find the critical value for \(t\) with \(\alpha = 0.1\) and degrees of freedom \(df = n - 1 = 23 - 1 = 22\).

Using the t-distribution table or a calculator for \(t_{0.1, 22}\), we find the critical value:
\[
t_{0.1, 22} \approx 1.321
\]

So, the rejection region for this right-tailed test is:
\[
t > 1.321
\]

### d) The decision is to:
- Since our calculated \(t\)-value \(2.646\) falls in the rejection region \(t > 1.321\), we **reject the null hypothesis** \(H_0\).

Therefore, the correct decision is:
\[
\text{reject } H_0 \text{ since the test statistic falls in the rejection region.}
\]

### Summary:
- **a)** Correct
- **b)** \( t = 2.646 \)
- **c)** \( t > 1.321 \)
- **d)** reject \( H_0 \)

Would you like more details or have any questions about this hypothesis test?

Here are 8 related questions for further exploration:
1. What is the formula for calculating the t-statistic in a one-sample t-test?
2. How do you determine the degrees of freedom in a t-test?
3. What is the significance level (\(\alpha\)) in hypothesis testing, and how is it used?
4. How do you interpret the p-value in the context of hypothesis testing?
5. What are the assumptions underlying the use of a t-test?
6. How does sample size affect the outcome of a t-test?
7. What is the difference between a one-tailed and a two-tailed test?
8. How do you use a t-distribution table to find critical values?

**Tip:** Always double-check your calculations and ensure you are using the correct formula and critical values for your hypothesis test.

Learn how to conduct a one-sample t-test for hypothesis testing using a sample of 23 observations with a mean of 57 and standard deviation of 3.44. Includes steps to calculate the test statistic, determine the rejection region at α = 0.1, and make a decision based on the test statistic.

Math Problem Statement

b) Calculate the test statistic, rounded to 3 decimal places. t

c) At α

Solution

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