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### Problems: The Central Limit Theorem Part 2

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11. A sample of size n=49 is taken from a population with a mean of µ = 100 and a standard deviation of σ = 14. Find the probability that the sample mean will lie between 96 and 104.

12. Suppose that the time it takes students at Gauss Normal School to walk from the campus to the movie theater is normally distributed with a mean equal to 15 minutes and a standard deviation of 3 minutes.

a) If we pick a student at random and have him make this walk, what is the probability that it will take him less than 19 minutes?
b) If we take a random sample of nine students, what is the probability that the total time spent walking will exceed 144 minutes?

13. Suppose that it is an established fact when 19-year-old males throw a baseball as far as they can, the results form a normal distribution with a mean of 180 feet and a standard deviation of 25 feet.

a. Suppose one nineteen-year-old male is chosen at random. What is the probability that he will throw over 190 feet?
b. Suppose that a sample of 25 nineteen-year-old males is chosen at random. What is the probability that the average throw of these 25 will be greater than 190 feet?
c. Suppose repeated random samplings of 100 19-year-old males are taken and the means of their throws computed. What percentage of the sample means would be greater than 190 feet?

14. Suppose it is established through comprehensive and exhaustive studies that when 50-to-60-year-old men run a 100 yards, their times average 24 seconds with a standard deviation of 5 seconds. Assume further that the distribution is normal.

a) What percentage of these men could run 100 yards in 20 seconds or less?
b) Suppose random samples of 16 men were tested and their run times averaged. What percentage of these averages would be 20 seconds or less?
c) Between what two times would we find the middle 95% of run times when we are looking at individual runners from this population of old guys?
d) Between what two times would we find the middle 95% of all sample means when the sample size is 100?

15. Suppose that a credit card company finds, using its computer records, that the average customer owes \$750 with a standard deviation of \$250. If a random sample of 225 customers is taken, what is the probability of getting a sample mean that is more than \$25 away from the true mean?

16. An average Quark battery lasts 20 hours with a standard deviation of 30 minutes.

a) Assuming a normal distribution, what percentage of the batteries will last more than 20 hours and 15 minutes?
b) If a sample of 16 is taken, what is the probability that the average lifetime will be greater than 20 hours and 15 minutes?

17. A sugar refiner produces bags of sugar that average 100 pounds with a standard deviation of .5 pounds. The distribution of bags is normally distributed.

a) What is the probability that one bag taken at random will weigh less than 99.6 pounds?
b) What is the probability that one bag taken at random will weigh 100.7 pounds or more?
c) What percentage of bags will weigh between 99.4 and 100.6 pounds?
c) How heavy must a bag be so that 90% of bags weigh less?
d) If a random sample of four bags is taken, what is the probability that the average weight will be less than 99.5 pounds?

If a random sample of nine bags is taken form this population, what is the probability that the sample average will be:

a) 99.5 pounds or less?
b) 100.3 pounds or more?
c) between 99.9 and 100.1 pounds?
d) How heavy must the sample average be so that fewer than 10% of sample averages would fall above it?

18. If we flip a coin and consider the number of heads we get, we have the following probability distribution:

 x P(x) 0 .5 1 .5

Compute the expected value and standard deviation of this distribution.

b) If we flip a coin twice and compute the fraction of the total that are heads, we get the following probability distribution:

 x P(x) 0 .25 .5 .5 1 .25

Compute the expected value and standard deviation of this distribution.

Flipping the coin three times gives the following probability distribution:

 x P(x) 0 1/8 1/3 3/8 2/3 3/8 1 1/8

Again, compute the expected value and standard deviation of this distribution.

Flipping the coin four times give the following:

 x P(x) 0 1/16 .25 4/16 .5 6/16 .75 4/16 1 1/16

Compute the expected value and standard deviation of this distribution.

Finally, flipping the coin five times gives this distribution:

 x P(x) 0 1/32 .2 5/32 .4 10/32 .6 10/32 .8 5/32 1 1/32

Compute the expected value and standard deviation of this distribution.

What happens to the mean as we flip the coin more times?

What happens to the standard deviation as we flip the coin more times?

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