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
48	67
23	51
31	52
37	54
45	55
50	25
23	38
31	34
53	44
55	68
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), we need more context, particularly for what type of test you are conducting. Generally, the type of test could be:

1. **t-test**: If you're comparing the means of two samples (such as comparing the mean scores of 30-second and 60-second ads).
2. **z-test**: If you're conducting a hypothesis test with a known population variance.
3. **chi-square test**: For categorical data comparison.
4. **ANOVA (F-test)**: If you're comparing more than two groups.

Given that you have two groups of data (30-second ads and 60-second ads), it's possible you're performing a **t-test** to compare the means of these two groups.

Here are the steps to find the critical value for a t-test:

1. **Determine the degrees of freedom (df):**
   \[
   df = n_1 + n_2 - 2
   \]
   where \(n_1\) and \(n_2\) are the number of observations in the 30-second and 60-second ads groups.

2. **Select the significance level (α)** for your test. This is typically 0.05 for a 95% confidence level, but the question might specify another value.

3. **Use a t-distribution table or calculator** to find the critical value based on the degrees of freedom and the significance level.

### Let's go step by step:

1. The sample sizes for both groups are:
   - For 30-second ads: \( n_1 = 10 \)
   - For 60-second ads: \( n_2 = 10 \)

   So, the degrees of freedom \( df = 10 + 10 - 2 = 18 \).

2. If we assume a significance level \( \alpha = 0.05 \), for a two-tailed test, the critical t-value can be found using a t-distribution table or a calculator for \( df = 18 \) and \( \alpha = 0.05 \).

Let me calculate the critical value for you.The critical value for a two-tailed t-test with 18 degrees of freedom and a significance level of \( \alpha = 0.05 \) is approximately **2.10**.

Would you like to proceed with additional calculations, or do you need more details about this? Here are five follow-up questions that might help you expand your understanding:

1. What are the sample means and standard deviations for the two groups (30-second and 60-second ads)?
2. How do you calculate the test statistic for comparing two independent means?
3. What is the p-value, and how does it relate to the critical value in hypothesis testing?
4. How would the critical value change if we used a one-tailed test instead of a two-tailed test?
5. How does increasing the sample size affect the critical value and the outcome of the test?

**Tip**: When conducting a t-test, always verify whether the variances of the two groups are equal. This determines whether you should use the pooled variance formula.

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: [48, 23, 31, 37, 45, 50, 23, 31, 53, 55]; 60-Second Ads: [67, 51, 52, 54, 55, 25, 38, 34, 44, 68]. Find the critical value(s).

Explore how to calculate critical values for a two-sample t-test comparing ad scores from 30-second and 60-second ads during a major championship event. Learn key concepts in hypothesis testing, such as degrees of freedom and significance levels, with step-by-step guidance. Suitable for high school and undergraduate students.

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