Math Problem Statement
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.)
Solution
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:
- t-test: If you're comparing the means of two samples (such as comparing the mean scores of 30-second and 60-second ads).
- z-test: If you're conducting a hypothesis test with a known population variance.
- chi-square test: For categorical data comparison.
- 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:
-
Determine the degrees of freedom (df): where and are the number of observations in the 30-second and 60-second ads groups.
-
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.
-
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:
-
The sample sizes for both groups are:
- For 30-second ads:
- For 60-second ads:
So, the degrees of freedom .
-
If we assume a significance level , for a two-tailed test, the critical t-value can be found using a t-distribution table or a calculator for and .
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 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:
- What are the sample means and standard deviations for the two groups (30-second and 60-second ads)?
- How do you calculate the test statistic for comparing two independent means?
- What is the p-value, and how does it relate to the critical value in hypothesis testing?
- How would the critical value change if we used a one-tailed test instead of a two-tailed test?
- 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.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
Two-Sample t-Test
Critical Values
Significance Level
Degrees of Freedom
Formulas
Degrees of freedom: df = n1 + n2 - 2
t-distribution critical value lookup based on df and significance level
Theorems
t-Test Critical Value Theorem
Suitable Grade Level
Grades 11-12, Undergraduate
Related Recommendation
T-Test for 30-Second vs 60-Second Ads: Critical Value Calculation
Find Critical Value for One-Sample T-Test with Significance Level α = 0.10 and Sample Size n = 20
Comparing Means of Two Populations Using Two-Sample t-Test
One-Tailed Hypothesis Test: Comparing Mean Scores for Two Test Versions
Two-Sample t-Test for Radio Ad Effectiveness