A simple random sample of size 

nequals=4040 

is drawn from a population. The sample mean is found to be 

107.5107.5​, 

and the sample standard deviation is found to be 

17.517.5. 

Is the population mean greater than 

100100 

at the 

alphaαequals=0.100.10 

level of​ significance?

Question content area bottom

Part 1

Determine the null and alternative hypotheses.

Upper H 0H0​: 

mu equals 100μ=100

Upper H 1H1​: 

mu greater than 100μ>100

Part 2

Compute the test statistic.

t 0t0

equals=enter your response here 

​(Round to two decimal places as​ needed.)

To determine if the population mean is greater than 100 at the \(\alpha = 0.10\) level of significance, we will follow these steps:

### Part 1: State the Hypotheses

The null hypothesis (H₀) and the alternative hypothesis (H₁) are:

- **Null Hypothesis (H₀)**: \(\mu = 100\)
- **Alternative Hypothesis (H₁)**: \(\mu > 100\)

This is a one-tailed test.

### Part 2: Compute the Test Statistic

We use the following formula for the test statistic \(t\):

\[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \]

Where:
- \(\bar{x} = 107.5\) (sample mean)
- \(\mu = 100\) (population mean under the null hypothesis)
- \(s = 17.5\) (sample standard deviation)
- \(n = 40\) (sample size)

Let's calculate the test statistic \(t\):

First, calculate the standard error of the mean (SEM):

\[ \text{SEM} = \frac{17.5}{\sqrt{40}} = \frac{17.5}{6.3246} \approx 2.767 \]

Now, calculate the test statistic \(t\):

\[ t = \frac{107.5 - 100}{2.767} \approx \frac{7.5}{2.767} \approx 2.71 \]

Thus, the test statistic \(t_0\) is approximately 2.71.

### Summary
- **Hypotheses**: \( H_0: \mu = 100 \), \( H_1: \mu > 100 \)
- **Test Statistic**: \( t_0 \approx 2.71 \)

Would you like more details or have any questions?

Here are five questions you can ask next:
1. How do we determine the critical value for a one-tailed t-test?
2. What is the p-value associated with our test statistic?
3. How do we interpret the results of this t-test?
4. What assumptions must be met to use a t-test in this scenario?
5. How can we calculate the confidence interval for the population mean?

**Tip:** Always verify that your sample size is sufficiently large to justify using the t-distribution, especially when dealing with small sample sizes.

Learn how to perform hypothesis testing to determine if the population mean is greater than 100 at a significance level of α = 0.10. Includes step-by-step calculation of the t-test statistic using sample data (sample mean = 107.5, sample standard deviation = 17.5, sample size = 40).

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