Question

what is the purpose of creating a randomization distribution (grouping data and randomizing it several times)?
the shaded area in the distribution below represents the probability that the original difference of means in any given problem is due to the grouping (randomness). would you say that the original difference of means here is caused by chance (randomness) or is it because of the treatment? why?
practice problems

1. a fitness organization is interested in the number of hours people exercise each week. they send out 40 volunteers to conduct research by collecting random samples of 25 people each and determining the number of hours that group exercised in the previous week. after looking at the sample means, the organization estimates that the mean number of hours exercised is 5.6 with a margin of error of 1.8. based on these values, what interval is likely to contain the true mean number of hours exercised in the previous week by the population?

Explanation:

Step1: Recall confidence - interval formula

The confidence interval for the population mean is given by $\bar{x}\pm E$, where $\bar{x}$ is the sample - mean and $E$ is the margin of error.

Step2: Identify values

We are given that $\bar{x} = 5.6$ (sample mean) and $E=1.8$ (margin of error).

Step3: Calculate the lower and upper bounds

The lower bound of the confidence interval is $\bar{x}-E=5.6 - 1.8=3.8$.
The upper bound of the confidence interval is $\bar{x}+E=5.6 + 1.8 = 7.4$.

Answer:

The interval is $[3.8,7.4]$.