11.2.2014, stats ch 7, probability

4. The distribution of sample means is not always a normal distribution. Under what circumstances is the distribution of sample means not normal? (11.2.2014, stats ch 7, probability )

4. The distribution of sample means will not be normal when it is based on small samples (n < 30) selected from a population that is not normal.

3. A sample is selected from a population with a mean of p = 40 and a standard deviation of o- = 8. a. If the sample has n = 4 scores, what is the expected value of M and the standard error of M? b. If the sample has n = 16 scores, what is the expected value of M and the standard error of M? (11.2.2014, stats ch 7, probability )

3. a. The expected value is μ = 40 and σM = 8/√4 = 4. b. The expected value is μ = 40 and σM = 8/√16 = 2.

1. Briefly define each of the following: a. Distribution of sample means b. Expected value of M c. Standard error of M (11.2.2014, stats ch 7, probability )

1. a. The distribution of sample means consists of the sample means for all the possible random samples of a specific size (n) from a specific population. b. The expected value of M is the mean of the distribution of sample means (μ). c. The standard error of M is the standard deviation of the distribution of sample means (σM = σ/n).

6. For a population with a mean of p = 70 and a standard deviation of o = 20, how much error, on average, would you expect between the sample mean (M) and the population mean for each of the following sample sizes? a. n = 4 scores b. n = 16 scores c. n = 25 scores (11.2.2014, stats ch 7, probability )

6. a. 20/4 = 10 points (box is square root) b. 20/16 = 5 points c. 20/25 = 4 points

10. For a population with a mean of p = 80 and a standard deviation of a = 12, find the z-score corresponding to each of the following samples. a. M = 83 for a sample of n = 4 scores b. M = 83 for a sample of n = 16 scores c. M = 83 for a sample of n = 36 scores (11.2.2014, stats ch 7, probability )

10. a. σM = 6 points and z = 0.50 b. σM = 3 points and z = 1.00 c. σM = 2 points and z = 1.50

11. A sample of n = 4 scores has a mean of M = 75. Find the z-score for this sample: a. If it was obtained from a population with p = 80 and if = 10. b. If it was obtained from a population with p = 80 and if = 20. c. If it was obtained from a population with p = 80 and if = 40. (11.2.2014, stats ch 7, probability )

11. a. σM = 5 points and z = 1.00 b. σM = 10 points and z = 0.50 c. σM = 20 points and z = 0.25

12. A population forms a normal distribution with a mean of p = 80 and a standard deviation of a = 15. For each of the following samples, compute the z-score the sample mean and determine whether the sample mean is a typical, representative value or an extreme value for a sample of this size. a. M = 84 for n = 9 scores b. M = 84 for n = 100 scores (11.2.2014, stats ch 7, probability )

12. a. With a standard error of 5, M = 84 corresponds to z = 0.80, which is not extreme. b. With a standard error of 1.5, M = 84 corresponds to z = 2.67, which is extreme.

13. A random sample is obtained from a normal population with a mean of p = 30 and a standard deviation of a = 8. The sample mean is M = 33. a. Is this a fairly typical sample mean or an extreme value for a sample of n = 4 scores? b. Is this a fairly typical sample mean or an extreme value for a sample of n = 64 scores? (11.2.2014, stats ch 7, probability )

13. a. With a standard error of 4, M = 33 corresponds to z = 0.75, which is not extreme. b. With a standard error of 1, M = 33 corresponds to z = 3.00, which is extreme.

a = 15. What is the probability of obtaining a sample mean greater than M = 97, a. for a random sample of n = 9 people? b. for a random sample of n = 25 people? (11.2.2014, stats ch 7, probability )

14. a. σM = 5, z = 0.60, and p = 0.7257 b. σM = 3, z = 1.00, and p = 0.8413

15. The scores on a standardized mathematics test for 8th-grade children in New York State form a normal distribution with a mean of p = 70 and a standard deviation of a = 10. a. What proportion of the students in the state have scores less than X = 75? b. If samples of n = 4 are selected from the population, what proportion of the samples will have means less than M = 75? c. If samples of n = 25 are selected from the population, what proportion of the samples will have means less than M = 75? (11.2.2014, stats ch 7, probability )

15. a. z = 0.50 and p = 0.6915 b. σM = 5, z = 1.00 and p = 0.8413 c. σM = 2, z = 2.50 and p = 0.9938

16. A population of scores forms a normal distribution with a mean of p = 40 and a standard deviation of o- = 12. a. What is the probability of randomly selecting a score less than X = 34? b. What is the probability of selecting a sample of n = 9 scores with a mean less than M = 34? c. What is the probability of selecting a sample of n = 36 scores with a mean less than M = 34? (11.2.2014, stats ch 7, probability )

16. a. z = -0.50 and p = 0.3085 b. σM = 4, z = -1.50 and p = 0.0668 c. σM = 2, z = -3.00 and p = 0.0013

17. A population of scores forms a normal distribution with a mean of p = 80 and a standard deviation of if = 10. a. What proportion of the scores have values between 75 and 85? b. For samples of n = 4, what proportion of the samples will have means between 75 and 85? c. For samples of n = 16, what proportion of the samples will have means between 75 and 85? (11.2.2014, stats ch 7, probability )

17. a. z = ±0.50 and p = 0.3830 b. σM = 5, z = ±1.00 and p = 0.6826 c. σM = 2.5, z = ±2.00 and p = 0.9544

18. At the end of the spring semester, the Dean of Students sent a survey to the entire freshman class. One question asked the students how much weight they had gained or lost since the beginning of the school year. The average was a gain of p = 9 pounds with a standard deviation of o- = 6. The distribution of scores was approximately normal. A sample of n = 4 students is selected and the average weight change is computed for the sample. a. What is the probability that the sample mean will be greater than M = 10 pounds? In symbols, what is p(M > 10)? b. Of all of the possible samples, what proportion will show an average weight loss? In symbols, what is p(M < 0)? c. What is the probability that the sample mean will be a gain of between M = 9 and M = 12 pounds? In symbols, what is p(9 < M < 12)? (11.2.2014, stats ch 7, probability )

18. a. z = 0.33 and p = 0.3707 b. z = 3.00 and p = 0.0013 c. p (0 < z < 1.00) = 0.3413

19. The machinery at a food-packing plant is able to put exactly 12 ounces of juice in every bottle. However, some items such as apples come in variable sizes so it is almost impossible to get exactly 3 pounds of apples in a bag labeled "3 lbs." Therefore, the machinery is set to put an average of p = 50 ounces (3 pounds and 2 ounces) in each bag. The distribution of bag weights is approximately normal with a standard deviation of a = 4 ounces. a. What is the probability of randomly picking a bag of apples that weighs less than 48 ounces (3 pounds)? b. What is the probability of randomly picking n = 4 bags of apples that have an average weight less than M = 48 ounces? (11.2.2014, stats ch 7, probability )

19. a. p(z < -0.50) = 0.3085 b. p(z < -1.00) = 0.1587

2. Describe the distribution of sample means (shape, expected value, and standard error) for samples of n = 36 selected from a population with a mean of p = 100 and a standard deviation of cr = 12.

2. The distribution of sample means will be normal (because n > 30), have an expected value of μ = 100, and a standard error of σM = 12/√36 = 2.

20. The average age for licensed drivers in the county is 1.1 = 40.3 years with a standard deviation of a = 13.2 years. a. A researcher obtained a random sample of n = parking tickets and computed an average age of M = 38.9 years for the drivers. Compute the z-score for the sample mean and find the probability of obtaining an average age this young or younger for a random sample of licensed drivers. Is it reasonable to conclude that this set of n = 16 people is a representative sample of licensed drivers? b. The same researcher obtained a random sample n = 36 speeding tickets and computed an average age of M = 36.2 years for the drivers. Compute z-score for the sample mean and find the probability of obtaining an average age this young or younger for a random sample of licensed drivers. Is it reasonable to conclude that this set of n = people is a representative sample of licensed drivers? (11.2.2014, stats ch 7, probability )

20. a. With a standard error of σM = 3.3, M = 38.9 corresponds to z = -0.42 and p = 0.3372. This is not an unusual sample. It is representative of the population. b. With a standard error of σM = 2.2, M = 36.2 corresponds to z = -1.86 and p = 0.0314. The sample mean is unusually small and not representative.

21. People are selected to serve on juries by randomly picking names from the list of registered voters. The average age for registered voters in the county is 1.1 = 44.3 years with a standard deviation of a = 12.4. A statistician computes the average age for a group of n = 12 people currently serving on a jury and obtains a mean of M = 48.9 years. a. How likely is it to obtain a random sample of n = 12 jurors with an average age equal to or greater than 48.9? b. Is it reasonable to conclude that this set of n = 12 people is not a representative random sample of registered voters? (11.2.2014, stats ch 7, probability )

21. a. With a standard error of 3.58 this sample mean corresponds to a z score of z = 1.28. A z score this large (or larger) has a probability of p = 0.1003. b. A sample mean this large should occur only 1 out of 10 times. This is not a very representative sample.

5. A population has a standard deviation of r = 30. a. On average, how much difference should exist between the population mean and the sample mean for n = 4 scores randomly selected from the population? b. On average, how much difference should exist for a sample of n = 25 scores? c. On average, how much difference should exist for a sample of n = 100 scores? (11.2.2014, stats ch 7, probability )

5. a. Standard error = 30/4 = 15 points b. Standard error = 30/25 = 6 points c. Standard error = 30/100 = 3 points

7. For a population with a standard deviation of a = 20, how large a sample is necessary to have a standard error that is: a. less than or equal to 5 points? b. less than or equal to 2 points? c. less than or equal to 1 point? (11.2.2014, stats ch 7, probability )

7. a. n ≥ 16 b. n ≥ 100 c. n ≥ 400

8. If the population standard deviation is a = 8, how large a sample is necessary to have a standard error that is: a. less than 4 points? b. less than 2 points? c. less than 1 point? (11.2.2014, stats ch 7, probability )

8. a. n > 4 b. n > 16 c. n > 64

9. For a sample of n = 25 scores, what is the value of the population standard deviation (cr) necessary to produce each of the following a standard error values? a. um = 10 points? b. cr,w = 5 points? c. crAi = 2 points? (11.2.2014, stats ch 7, probability )

9. a. σ = 50 b. σ = 25 c. σ = 10

A population forms a normal distribution with a mean of p = 60 and a standard deviation of a = 12. For a sample of n = 36 scores from this population, what is the probability of obtaining a sample mean greater than 64? p(M > 64) = ? (11.2.2014, stats ch 7, probability )

STEP 1 Rephrase the probability question as a proportion question. Out of all of the possible sample means for n = 36, what proportion has values greater than 64? All of the possible sample means is simply the distribution of sample means, which is normal, with a mean of p = 60 and a standard error of a 12 12 0-m = VW .\/ 6 = = =2 The distribution is shown in Figure 7.13(a). Because the problem is asking for the proportion greater than M = 64, this portion of the distribution is shaded in Figure 7.13(b). STEP 2 Compute the z-score for the sample mean. to a z-score of M — 64 — 60 4 z= = =2.00 QM 2 2 Therefore, p(M > 64) = p(z > 2.00) A sample mean of M = 64 corresponds STEP 3 Look up the proportion in the unit normal table. Find z = 2.00 in column read across the row to find p = 0.0228 in column C. This is the answer as shown Figure 7.13(c). p(M > 64) = p(z> 2.00) = 0.0228 (or 2.28%)