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Sampling Distributions

1. Given the discrete uniform population

f(x) = 1/3, x = 2, 4, 6,
0 elsewhere,

Find the probability that a random sample of size 54, selected with replacement, will yield a sample mean greater than 4.1 but less than 4.4. Assume the means to be measured to the nearest tenth.

2. If the standard deviation of the mean for the sampling distribution of random samples of size 36 from a large or infinite population is 2, how large must the size of the sample become if the standard deviation is to be reduced to 1.2?

3. The heights of 1000 students are approximately normally distributed with a mean of 174.5 centimeters and a standard deviation of 6.9 centimeters. If 200 random samples of size 25 are drawn from this population and the means recorded to the nearest tenth of a centimeter, determine

a) the mean and standard deviation of the sampling distribution of X;
b) the number of sample means that fall between 172.5 and 175.8 centimeters inclusive;
c) the number of sample means falling below 172.0 centimeters

4. The amount of time that a drive-through bank teller spends on a customer is a random variable with a mean µ= 3.2 minutes and a standard deviation ơ = 1.6 minutes. If a random sample of 64 customers is observed, find the probability that their mean time at the teller's counter is

a) at most 2.7 minutes;
b) more than 3.5 minutes;
c) at least 3.2 minutes but less than 3.4 minutes.

5. A random sample of size 25 is taken from a normal population having a mean of 80 and a standard deviation of 5. A second random sample of size 36 is taken from a different normal population having a mean of 75 and a standard deviation of 3. Find the probability that the sample mean computed from the 25 measurements will exceed the sample mean computed from the 36 measurements by at least 3.4 but less than 5.9. Assume the difference of the means to be measured to the nearest tenth.

6. The mean score for freshmen on an aptitude test at a certain college is 540, with a standard deviation of 50. What is the probability that two groups of students selected at random, consisting of 32 and 50 students, respectively, will differ in their mean scores by

a) more than 20 points?
b) an amount between 5 and 10 points? Assume the means to be measured to any degree of accuracy.

Solution Preview

1. Given the discrete uniform population

f(x) = 1/3, x = 2, 4, 6,
0 elsewhere,

Find the probability that a random sample of size 54, selected with replacement, will yield a sample mean greater than 4.1 but less than 4.4. Assume the means to be measured to the nearest tenth.

The population mean is 2(1/3) + 4(1/3) + 6(1/3) = 4. The population standard deviation is calculated by:

So, σ = √(1/3)[4 + 0 + 4] = √(8/3) = 1.633.

Now that we know the population mean and standard deviation, we can find the z-scores for x = 4.1 and x = 4.4, and use those numbers to find the probability that the sample mean is greater than 4.1 and less than 4.4.

The z statistic is calculated as

Where SE is the standard error and is equal to

x = 4.1:

z = 4.1 - 4 = 0.1 = 0.45
1.633/√54 0.222

x = 4.4:

z = 4.4 - 4 = 0.4 = 1.8
1.633/√54 0.222

If you look at a z-distribution table, the area between z = 0 and z = 0.45 is 0.1736, and the area between z = 0 and z = 1.80 is 0.4641. Therefore, the area between z = 0.45 and z = 1.80 is 0.4641 - 0.1736 = 0.2905.

The probability that the mean is between 4.1 and 4.4 is 29.05%.

2. If the standard deviation of the mean for the sampling distribution of random samples of size 36 from a large or infinite population is 2, how large must the size of the sample become if the standard deviation is to be reduced to 1.2?

The formula for the standard deviation of the sampling distribution ...

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