You also can find the t-statistic on StatCrunch. If you are given summary statistics, go to
StatT Stats Two Sample > With Summary.
To test if two populations have different means, a random sample of 74 people from the first population had a mean of 17.3 standard deviation of 4.1. A random sample of 83 people from the second population had a mean of 19.2 and standard deviation of 3.7. Use these summary statistics to compute t and the p-value on StatCrunch. Round to four decimal places.
t
p-value-
The Correct Answer and Explanation is :
To determine whether two populations have different means, we can perform a two-sample t-test using the provided summary statistics. Here’s how to compute the t-statistic and p-value:
Given Data:
- Sample 1:
- Sample size ((n_1)): 74
- Sample mean ((\overline{x}_1)): 17.3
- Sample standard deviation ((s_1)): 4.1
- Sample 2:
- Sample size ((n_2)): 83
- Sample mean ((\overline{x}_2)): 19.2
- Sample standard deviation ((s_2)): 3.7
Step 1: Calculate the Pooled Standard Deviation
The pooled standard deviation ((s_p)) is calculated using the formula:
[ s_p = \sqrt{\frac{(n_1 – 1) \cdot s_1^2 + (n_2 – 1) \cdot s_2^2}{n_1 + n_2 – 2}} ]
Substituting the given values:
[ s_p = \sqrt{\frac{(74 – 1) \cdot 4.1^2 + (83 – 1) \cdot 3.7^2}{74 + 83 – 2} ]
[ s_p = \sqrt{\frac{73 \cdot 16.81 + 82 \cdot 13.69}{155}} ]
[ s_p = \sqrt{\frac{1222.33 + 1125.48}{155}} ]
[ s_p = \sqrt{\frac{2347.81}{155}} ]
[ s_p \approx \sqrt{15.14} ]
[ s_p \approx 3.89 ]
Step 2: Calculate the Standard Error of the Difference Between Means
The standard error ((SE)) of the difference between the two sample means is:
[ SE = s_p \cdot \sqrt{\frac{1}{n_1} + \frac{1}{n_2}} ]
Substituting the values:
[ SE = 3.89 \cdot \sqrt{\frac{1}{74} + \frac{1}{83}} ]
[ SE = 3.89 \cdot \sqrt{0.01351 + 0.01205} ]
[ SE = 3.89 \cdot \sqrt{0.02556} ]
[ SE \approx 3.89 \cdot 0.160 ]
[ SE \approx 0.623 ]
Step 3: Calculate the t-Statistic
The t-statistic ((t)) is calculated as:
[ t = \frac{\overline{x}_1 – \overline{x}_2}{SE} ]
Substituting the values:
[ t = \frac{17.3 – 19.2}{0.623} ]
[ t = \frac{-1.9}{0.623} ]
[ t \approx -3.05 ]
Step 4: Determine the Degrees of Freedom
The degrees of freedom ((df)) for the t-test is:
[ df = n_1 + n_2 – 2 ]
[ df = 74 + 83 – 2 ]
[ df = 155 ]
Step 5: Find the p-Value
Using a t-distribution table or statistical software with (df = 155) and (t = -3.05), we find the p-value. For a two-tailed test, the p-value is approximately 0.0025.
Conclusion:
Since the p-value (0.0025) is less than the common significance level of 0.05, we reject the null hypothesis. This indicates that there is a statistically significant difference between the means of the two populations.
Using StatCrunch:
To perform this analysis in StatCrunch:
- Navigate to Stat > T Stats > Two Sample > With Summary.
- Enter the sample sizes, means, and standard deviations for both samples.
- Select the appropriate hypothesis test (e.g., two-tailed).
- Click Compute to obtain the t-statistic and p-value.
For a visual demonstration, you can refer to the following tutorial:
videoTwo Sample t Test in StatCrunchturn0search1