An annual championship event is the most widely watched sporting event in a country each year. In recent​ years, there has been a great deal of interest in the ads that appear during the game. These ads vary in length with most lasting 30 seconds or 60 seconds. The accompanying data represent the ad length and ad scores from a recent championship. Complete parts​ (a) through​ (c).

30-Second Ads	60-Second Ads

41	68

24	52

39	56

38	50

43	56

50	29

21	37

40	33

59	44

56	70
Find the critical​ value(s).

enter your response here

​(Use a comma to separate answers as needed. Round to two decimal places as​ needed.)

To find the critical value(s) in this context, it seems you're asking for a hypothesis test comparing the means of the two groups: 30-second ads and 60-second ads. Typically, the critical value(s) depend on:

1. The type of hypothesis test (e.g., two-tailed or one-tailed).
2. The level of significance (α), which is often 0.05.
3. The degrees of freedom for the test.

Let's assume we are conducting a **two-tailed t-test** to compare the means of the two independent samples (30-second ads and 60-second ads). The steps are:

### Step 1: Determine the degrees of freedom (df)
For two independent samples, the degrees of freedom can be calculated using this formula:

\[
df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{ \left( \frac{s_1^2}{n_1} \right)^2 }{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}}
\]

where:
- \(s_1^2\) and \(s_2^2\) are the sample variances of the 30-second and 60-second ads, respectively.
- \(n_1\) and \(n_2\) are the sample sizes of the two groups.

Let's calculate the degrees of freedom and find the critical value from the t-distribution.

I will start by calculating the sample variances and then proceed with the calculation.The degrees of freedom for the t-test are approximately **17.71**. Using a significance level of 0.05 for a two-tailed test, the critical t-value is approximately **2.10**.

### Summary:
- Degrees of freedom: 17.71
- Critical t-value (two-tailed, α = 0.05): 2.10

Would you like further details on how this applies to the rest of your analysis or any additional clarifications?

Here are 5 related questions for further exploration:
1. What is the null hypothesis in a two-sample t-test?
2. How do you interpret the p-value in this context?
3. What would be the critical value if you used a one-tailed test?
4. How do sample size and variance affect the degrees of freedom?
5. How is the confidence interval related to the t-test result?

**Tip:** Always check the assumptions of normality and equal variances when conducting t-tests.

An annual championship event is the most widely watched sporting event in a country each year. In recent years, there has been a great deal of interest in the ads that appear during the game. These ads vary in length with most lasting 30 seconds or 60 seconds. The accompanying data represent the ad length and ad scores from a recent championship. Complete parts (a) through (c).

30-Second Ads: 41, 24, 39, 38, 43, 50, 21, 40, 59, 56
60-Second Ads: 68, 52, 56, 50, 56, 29, 37, 33, 44, 70
Find the critical value(s).

Learn how to calculate the critical t-value for a two-sample t-test comparing the effectiveness of 30-second and 60-second ads. This statistical problem involves hypothesis testing, variance analysis, and calculating degrees of freedom. Suitable for undergraduate students studying introductory statistics.

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