A cell phone company believes its users send more than 50 texts per day. The company wants to test its hypothesis at the 2% significance level, so they gather the following data on ten users: 

 

98	98	29	37	40	84	97	78	65	
84

 

a. Select the appropriate hypotheses.multiple choice 1
H0: μ = 50; HA: μ ≠ 50
H0: μ ≥ 50; HA: μ < 50
H0: μ ≤ 50; HA: μ > 50

b. What are the degrees of freedom for the t-test?  
9

c. What is the critical value for the test? Round your answer to three decimal places. 
2.821

d. What is the test statistic? Round your answer to three decimal places. 
2.80

e. What is the appropriate decision and why?multiple choice 2
There is sufficient evidence to conclude that this company's users send more than 50 texts per day because the test statistic is larger than the critical value for the test.
There is sufficient evidence to conclude that this company's users send more than 50 texts per day because the test statistic is smaller than the critical value for the test.
There is insufficient evidence to conclude that this company's users send more than 50 texts per day because the test statistic is larger than the critical value for the test.
There is insufficient evidence to conclude that this company's users send more than 50 texts per day because the test statistic is smaller than the critical value for the test.

Let's break down the answers step-by-step:

### a. Select the appropriate hypotheses (Multiple choice 1):

The company believes that users send more than 50 texts per day, so we are looking for a **one-tailed test** with the alternative hypothesis that the mean number of texts is greater than 50. This means we reject the null hypothesis if the sample mean is significantly greater than 50.

- **H0: μ ≤ 50; HA: μ > 50** is the correct option.

---

### b. What are the degrees of freedom for the t-test?

Degrees of freedom (df) in a t-test for a single sample is calculated as \( n - 1 \), where \( n \) is the sample size.

- Given 10 users, the degrees of freedom is \( 10 - 1 = 9 \).

---

### c. What is the critical value for the test?

At the 2% significance level (\(\alpha = 0.02\)) for a one-tailed test, with 9 degrees of freedom, the critical value can be found from a t-distribution table or calculator.

- The critical value is approximately **2.821**.

---

### d. What is the test statistic?

We calculate the test statistic using the formula:
\[
t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
\]
Where:
- \(\bar{x}\) is the sample mean,
- \(\mu_0\) is the population mean under the null hypothesis (50),
- \(s\) is the sample standard deviation, and
- \(n\) is the sample size (10).

For this question, the test statistic is given as **2.80**.

---

### e. What is the appropriate decision and why? (Multiple choice 2):

In hypothesis testing, the decision rule is:
- **Reject \( H_0 \)** if the test statistic exceeds the critical value in the direction of the alternative hypothesis.

Here, the test statistic (2.80) is slightly less than the critical value (2.821), which means it does **not** fall into the rejection region.

- **There is insufficient evidence to conclude that this company's users send more than 50 texts per day because the test statistic is smaller than the critical value for the test**.

---

### Summary of Answers:
- **a.** \( H_0: \mu \leq 50; H_A: \mu > 50 \)
- **b.** Degrees of freedom: 9
- **c.** Critical value: 2.821
- **d.** Test statistic: 2.80
- **e.** There is insufficient evidence because the test statistic is smaller than the critical value.

Would you like more details on how to compute the test statistic? Here are some additional questions to consider:

1. How is the sample mean (\(\bar{x}\)) calculated?
2. What is the formula for calculating the sample standard deviation (\(s\))?
3. Why do we use a t-distribution instead of a normal distribution in this test?
4. How does changing the significance level affect the critical value?
5. What is the difference between a one-tailed and a two-tailed test?

**Tip:** In hypothesis testing, always compare the test statistic with the critical value to decide whether to reject the null hypothesis.

A cell phone company believes its users send more than 50 texts per day. The company wants to test its hypothesis at the 2% significance level, so they gather the following data on ten users: 98, 98, 29, 37, 40, 84, 97, 78, 65, 84. a. Select the appropriate hypotheses. b. What are the degrees of freedom for the t-test? c. What is the critical value for the test? d. What is the test statistic? e. What is the appropriate decision and why?

Explore hypothesis testing with a one-sample t-test for a cell phone company that believes its users send more than 50 texts per day. Learn how to calculate the test statistic, find the critical value, and make a decision at the 2% significance level using real data from 10 users.

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