Show all work Question 2 a Big town Fisheries recently stock

Show all work

Question 2.

(a). Big town Fisheries recently stocked a new lake in a city with 2,000 fish of various sizes. The distribution of the lengths of these fish is approximately normal.

(i). Big town Fisheries claims that the mean length of the fish is 8 inches. If the claim is true, which of the following would be more likely?

• A random sample of 15 fish having a mean length that is greater than 10 inches or

• A random sample of 50 fish having a mean length that is greater than 10 inches

Justify your answer in terms of the Central Limit Theorem. (5 points)

(ii). Suppose the standard deviation of the sampling distribution of the sample mean for random samples of size 50 is 0.3 inch. If the mean length of the fish is 8 inches, use the normal distribution to compute the probability that a random sample of 50 fish will have a mean length less than 7.5 inches.

(5 points)

(iii). Even if the population of interest for the fish was non-normal, do you think the calculation in part (ii) would still be reasonable? (5 points)

(b). A local radio station plays 40 rock-and-roll songs during each 4-hour show. The program director at the station needs to know the total amount of airtime for the 40 songs so that time can be programmed during the show for news and advertisement. The distribution of the lengths of rock-and-roll songs, in minutes, is roughly symmetric with a mean length of 3.9 minutes and a standard deviation of 1.1 minutes.

(i). Describe the sampling distribution of the sample mean song lengths for random samples of 40 rock-and-roll songs. (5 points)

(ii). If the program manager schedules 80 minutes of news and advertisement for 4-hours(240 minute) show, only 160 minutes are available for music. Approximately what is the probability that the total amount of time needed to play 40 randomly selected rock-and-roll songs exceeds the available airtime?

(5 points)

(c). A tire manufacturer designed a new tread pattern for its all-weather tires. Repeated tests were conducted on cars of approximately the same weight travelling at 60 miles per hour. The test showed that the new tread pattern enables the cars to stop completely in an average distance of 125 feet with a standard deviation of 6.5 feet and that the stopping distances are approximately normally distributed.

(i). What is the 70th percentile of the distribution of stopping distances? (5 points)

(ii).What is the probability that at least 2 cars out of 5 randomly selected cars in the study will stop in a distance that is greater than the distance calculated in part (i)? (5 points)

(iii). What is the probability that a randomly selected sample of 5 cars in the study will have a mean stopping distance of at least 130 feet? (5 points)

Solution

a)

i)

OPTION A: A random sample of 15 fish having a mean length that is greater than 10 inches

This is because the standard error here is s/sqrt(n). Thus, there is greater standard error for smaller n, so it makes it more probable to have a result that is of some distance from the mean.

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ii)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    7.5      
u = mean =    8      
          
s = standard deviation =    0.3      
          
Thus,          
          
z = (x - u) / s =    -1.666666667      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z <   -1.666666667   ) =    0.047790352 [ANSWER]

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iii)

Yes, because n = 50, and by central limit theorem, the distrbution of sample means will be approximately normally distributed for a large number of iterates.

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