Answer :
a) The shape of the distribution is approximately normal.
b) The mean of the sampling distribution is of: 25 minutes.
c) The 10% condition is met, as both samples are of 30 observations, and the standard deviation is of: 1.57 minutes.
d) The probability that the difference in sample means is less than 20 minutes is of: 0.0007 = 0.07%.
How to obtain probabilities using the normal distribution?
The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- The z-score represents how many standard deviations the measure X is above or below the mean of the distribution, depending if the obtained z-score is positive or negative.
- Using the z-score table, the p-value associated with the calculated z-score is found, and it represents the percentile of the measure X in the distribution.
- By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex], as long as the sample size is greater than 30, meeting the 10% condition.
Applying the Central Limit Theorem, the shape of the distribution of differences is approximately normal.
The standard error for each sample is given as follows:
- TR: 7/square root of 30 = 1.28.
- MWF: 5/square root of 30 = 0.91.
Hence the mean and the standard deviation for the distribution of differences is given as follows:
- Mean: 75 - 50 = 25.
- Standard deviation: square root(1.28² + 0.91²) = 1.57.
The probability that the difference in sample means is less than 20 minutes is the p-value of Z when X = 20, hence:
Z = (20 - 25)/1.57
Z = -3.18
Z = -3.18 has a p-value of 0.0007.
More can be learned about the normal distribution at https://brainly.com/question/25800303
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