Question

An observational study of teams fishing for the red spiny lobster in a certain body of water was conducted and the results published in a science magazine. One of the variables of interest was the average distance separating trapsminuscalled ​"trap ​spacing"minusdeployed by the same team of fishermen. Trap spacing measurements​ (in meters) for a sample of seven teams of fishermen are shown in the accompanying table. Of interest is the mean trap spacing for the population of red spiny lobster fishermen fishing in this body of water. Complete parts below:

data: 94 98 104 94 80 71 88

a. Identify the target parameter for this study. The target parameter for this study is mu .

b. Compute a point estimate of the target parameter. ​(Round to two decimal places as​ needed.)

c. What is the problem with using the normal​ (z) statistic to find a confidence interval for the target​ parameter? A. The point estimate is too large to determine an accurate​ z-statistic. B. The​ z-statistic is used for confidence intervals for​ proportions, not means. C. The point estimate is too large to determine an accurate critical value. D. The sample is small and the trap spacing population has unknown distribution and standard deviation.

d. Find a​ 95% confidence interval for the target parameter. left parenthesis nothing comma nothing right parenthesis (Round to one decimal place as​ needed.)

e. Give a practical interpretation of the​ interval, part d. Choose the correct answer below. A. There is a​ 95% probability that the true mean trap spacing distance is the mean of the interval. B. One can be​ 95% confident the true mean trap spacing distance is one of the end points of the above interval. C. One can be​ 95% confident the true mean trap spacing distance lies at the mean of the above interval. D. One can be​ 95% confident the true mean trap spacing distance lies within the above interval.

f. What conditions must be satisfied for the​ interval, part d​, to be​ valid? Select all that apply. A. The sample is randomly selected from the population. B. The sample has a relative frequency distribution that is approximately normal. C. The sample must be large enough that the Central Limit Theorem applies. D. The population has a relative frequency distribution that is approximately normal.

Answer 1

a.

The target parameter for this study is the mean trap spacing for the population of red spiny lobster fishermen fishing in this body of water.

b.

Point estimate of the target parameter = = (94 + 98 + 104 + 94 + 80 + 71 + 88) /7 = 89.86

c.

To use the the normal​ (z) statistic to find a confidence interval, the sample size must be at least 30 to approximate the sampling distribution of mean as normal distribution.

D. The sample is small and the trap spacing population has unknown distribution and standard deviation.

d.

Sample variance, S^2 = (94 - 89.86)^2 + (98 - 89.86)^2 + (104 - 89.86)^2 + (94 - 89.86)^2 + (80 - 89.86)^2 + (71 - 89.86)^2 + (88 - 89.86)^2 ] /(7-1) = 126.1429

Standard deviation , S = = 11.23133

Standard error of mean, SE = S / = 11.23133 / = 4.245044

Degree of freedom = n-1 = 7-1 = 6

Critical value of t for 95% confidence interval and df = 6 is  2.447

95% confidence interval of true mean is,

(89.86 - 2.447 * 4.245044 ,  89.86 + 2.447 * 4.245044)

(79.47 ,  100.25)

e.

The practical interpretation of the​ interval is,

D. One can be​ 95% confident the true mean trap spacing distance lies within the above interval.

f.

The conditions to be be satisfied for the​ interval in part d​, to be​ valid are,

A. The sample is randomly selected from the population.

C. The sample must be large enough that the Central Limit Theorem applies.

D. The population has a relative frequency distribution that is approximately normal.