We are interested in assessing the number of ropes a person can jump rope. We have a random sample of 30 people and find that the average person can jump rope 6.4 times with a standard deviation of 3.6. What is the $98 \%$ confidence interval for the actual average number of ropes a person can jump? Enter your answers accurate to one decimal place. Can we conclude that the actual average number of ropes a person can jump is at most 7.4? Yes No Not enough information to determine this Beyond representative sampling, what assumptions are needed to satisfy the requirements for this confidence interval? $X \sim$ binomial and at least 5 successes and failures $X \sim$ binomial, $n p \geq 5$ and $n q \geq 5$ $X \sim$ normal or $n \geq 30$ $X \sim$ normal and $n \geq 30$ Submit Question

Step 1 ：We are given a random sample of 30 people and find that the average person can jump rope 6.4 times with a standard deviation of 3.6. We are asked to find the 98% confidence interval for the actual average number of ropes a person can jump.

Step 2 ：We use the formula for the confidence interval, which is \(\bar{X} \pm Z \frac{\sigma}{\sqrt{n}}\), where \(\bar{X}\) is the sample mean, \(Z\) is the Z-score corresponding to the desired confidence level, \(\sigma\) is the standard deviation, and \(n\) is the sample size.

Step 3 ：Substituting the given values into the formula, we get \(6.4 \pm 2.33 \frac{3.6}{\sqrt{30}}\).

Step 4 ：Calculating the standard error, which is \(\frac{\sigma}{\sqrt{n}}\), we get approximately 0.657.

Step 5 ：Substituting the standard error into the formula, we get \(6.4 \pm 2.33 \times 0.657\).

Step 6 ：Calculating the confidence interval, we get approximately \((4.9, 7.9)\).

Step 7 ：The 98% confidence interval for the actual average number of ropes a person can jump is approximately \(\boxed{(4.9, 7.9)}\).

Step 8 ：Since 7.4 is within this interval, we cannot conclude that the actual average number of ropes a person can jump is at most 7.4.

Step 9 ：For the third question, the assumption needed for this confidence interval is that the sample size is large (\(n \geq 30\)) and the data is normally distributed or approximately normal. This is because the Central Limit Theorem states that if the sample size is large enough, the sampling distribution of the mean will be approximately normal regardless of the shape of the population distribution.