(A) Incorrect! A parameter is a number that describes the population. (C) Incorrect! In a Random Sample, not just a sample.

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1 AP Statistics - Problem Drill 15: Sampling Distributions No. 1 of 10 Instructions: (1) Read the problem statement and answer choices carefully (2) Work the problems on paper 1. Which one of the following is correct? (A) Parameter is a number that describes the sample. (B) Statistics is a number that describes the sample. (C) In a sample every unit in the population has equal probability of being sampled. (D) The probability distribution of a parameter is called its sampling distribution. (E) Parameters describe samples and statistics describe populations. A parameter is a number that describes the population. (B) Correct This is correct because it is statistics, not parameter that describes the sample. In a Random Sample, not just a sample. The probability distribution of a Statistics is called its sampling distribution. Parameters are numerical representations of populations while statistics are numbers that describe the sample. This question is on the definition of statistics and population. It is statistics, not parameter that describes the sample. (B) Statistics is a number that describes the sample.

2 No. 2 of Compute the sampling distribution for three tosses of a fair coin; X = 1 for heads, and X = 0 for tails, which of the following is wrong? (A) The possible outcomes of combinations are 000, 001, 010, 100, 110, 101, 011, 111. (B) The possible outcomes for the s are 0, 3, 2/3, 1 (C) P( =0)=8 (D) P( =2/3)=2/8 (E) P( =3)=3/8 All the possible outcomes of combinations are 000,001,010,100,110,101,011,111, this is not wrong. The possible outcomes for the s are 0, 3, 2/3, 1, this is not wrong. P( =0)=8, this is not wrong. Because P( =2/3)=3/8 (they are 110, 101, 011, 3 out of 8), so 2/8 is wrong, We should choose this one. P( =3)=3/8 is not wrong. P( =0)=8 (000) P( =3)=3/8 (100,010,001) P( =2/3)=3/8 (110,101,011) P( =1)=8 (111) All the possible outcomes of combinations are 000,001,010,100,110,101,011,111, this is not wrong. The possible outcomes for the s are 0, 3, 2/3, 1, this is not wrong. P( =0)=8, this is not wrong. Because P( =2/3)=3/8 (they are 110, 101, 011, 3 out of 8), so 2/8 is wrong, We should choose this one. P( =3)=3/8 is not wrong.

3 No. 3 of Compute the sampling distribution for three tosses of an unfair coin; X = 1 for heads, and X = 0 for tails, p(x=1)=0.2, p(x=0)=0.8, which of the following is wrong? (A) The possible outcomes of combinations are 000, 001, 010, 100, 110, 101, 011, 111 (B) The possible outcomes for the s are 0/3, 3, 2/3, 3/3 (C) P( =0)=0.8 3 (D) P( =2/3)= 0.2x0.8 2 (E) P( =3)=3* 0.2x0.8 2 All the possible outcomes of combinations are 000,001,010,100,110,101,011,111, this is not wrong. The possible outcomes for the s are 0, 3, 2/3, 1, this is not wrong. P( =0)= 0.8 3, this is not wrong. Because P( =2/3)= 0.8x0.2 2 (they are 110, 101, 011) The probability associated 3 with this is C() = 3(.04)(.8) =.06 E) Incorrect Because P( =3)=3* 0.2x0.8 2 the probability associated with this is: 3 C() = 3(.2)(.8) 2 =.384 P( =0)= 1* (000) P( =3)= 3* 0.2x0.8 2 (100,010,001) P( =2/3)= 3* 0.8x0.2 2 (110,101,011) P( =1)=1* (111) All the possible outcomes of combinations are 000,001,010,100,110,101,011,111, this is not wrong The possible outcomes for the s are 0, 3, 2/3, 1, this is not wrong. P( =0)= 0.8 3, this is not wrong. Because P( =2/3)= 3*0.8x0.2 2 (they are 110, 101, 011) There are 3 ways to create two heads and one tail. Take the probability times 3 to get the result.. Because P( =3)=3* 0.2x0.8 2 (D) P( =2/3)= 0.2x0.8 2

4 No. 4 of If X is a random variable with the probability p(x=0)=3,p(x=1)=3, P(X=2)=3, calculate μ (i.e. the expected value of X). (A) μ=0 (B) μ=1 (C) μ=2 (D) μ=3 (E) μ=3 μ= =1 (B) Correct! μ= =1 μ= =1 μ= =1 μ= =1 μ=p(x=0) 3+p(X=1) 3+p(X=2) 3= =1, therefore only (B) is correct, (A)(C)(D)(E) are wrong. (B) μ=1

5 No. 5 of If X is a random variable with the probability p(x=0)=3, p(x=1)=3, P(X=2)=3, then for the sampling distribution of the sample mean for samples of size n = 2, which one of the following is wrong? (A) The possible outcomes of combinations are 00, 01, 02, 10, 11, 12, 20, 21, 22 (B) p( =0.5)=2/ (C) p( =1)=2/ (D) p( =2)= (E) p( =0)= The possible outcomes of combinations are 00, 01, 02, 10, 11, 12, 20, 21, 22, so this is not wrong. p( =0.5)=2/, this is not wrong. (C) Correct! p( =1)=3/, because 02,11,20 are the three possible outcomes out of, so it is 3/ not 2/, this is wrong, so we choose this one. p( =2)=, this is not wrong. P( =0)=, this is not wrong. Sample 0, 0 0, 1 0, 2 1, 0 1, 1 1, 2 2, 0 2, 1 2, P This gives the following sampling distribution: P( = ) 2/ 3/ 2/ P( =0)= (00) P( =0.5)= 2/ (01,10) P( =1)= 3/ (20,02,11) P( =1.5)= 2/ (12,21) Therefore only (C) is chosen because it s the only wrong answer. (C) p( =1)=2/

6 No. 6 of Which of the following is NOT a true statement about the sampling distribution of the mean, created from a normal population? (A) The sampling distribution of sample means is normally distributed. (B) The mean of the sampling distribution = mean of the population. (C) Samples can be any size. (D) The spread of the sampling distribution = the spread of the population. (E) The standard deviation of the sample is approximately equal to the standard deviation of the population. This is a correct statement about the sampling distribution. This is a correct statement about the sampling distribution. This is a correct statement about the sampling distribution. This is an incorrect statement. The spread of the sampling distribution equals the spread of the population divided by the square root of the sample size.. This is a correct statement about the sampling distribution. The sampling distribution is a distribution of means. These means are more closely centered on the true mean of the population then individual terms. The spread of the sampling distribution is found by taking the standard deviation of the population and dividing it by the square root of the sample size. (D) The spread of the sampling distribution = the spread of the population.

7 No. 7 of You are interested in the general public's attitude to a global warming issue. Which of the following would return a simple random sample? (A) A survey in a university. (B) Sample a number randomly selected households at various times during the week. (C) A telephone survey is conducted on weekday mornings. Using a random number generator to select household phone numbers. (D) A survey in Wal-Mart. (E) A survey of your friends and family. A university survey cannot represent a general public s attitude. (B) Correct! Sample a number randomly selected households at various times during the week. This is correct because house is randomly selected and time is also randomly selected. Survey is conducted only on mornings; it should be on any time. Wal-Mart survey cannot represent general public.. Asking your friends and family is not a random sample. The principle of random sample must be that each element in the sample has equal opportunity of be selected; otherwise it cannot be regarded as random sample. A university survey cannot represent a general public s attitude. (B) Correct! Sample a number randomly selected households at various times during the week. This is correct because house is randomly selected and time is also randomly selected. Survey is conducted only on mornings; it should be on any time. Wal-Mart survey cannot represent general public.. Surveying friends and family is not random. (B) Sample a number randomly selected households at various times during the week.

8 No. 8 of Large samples (n>30) are taken from a population that is not normal. Which of the following is false? (A) The center of the distribution is very close to the true population mean. (B) The spread of the sample means will be the same as the spread of the population data. (C) The distribution of sample means is approximately normal. (D) There are no outliers. (E) The standard deviation of the sample is approximately equal to the standard deviation of the population. The centre of the distribution is very close to the true population mean, this is correct, so we don t choose it. (B) Correct! The spread of the sample means will be smaller than, not the same as, the spread of the population data So this one is false. The distribution of sample means is approximately normal, this is correct, so we don t choose it. There are no outliers, this is correct, so we don t choose it.! This is a true statement concerning sampling distributions. (A). Incorrect! The centre of the distribution is very close to the true population mean, this is correct, so we don t choose it. (B). Correct! The spread of the sample means will be smaller than, not the same as, the spread of the population data So this one is false. (C). Incorrect! The distribution of sample means is approximately normal, this is correct, so we don t choose it. (D). Incorrect! There are no outliers, this is correct, so we don t choose it. (E)Incorrect! This is a true statement concerning sampling distributions. (B) The spread of the sample means will be the same as the spread of the population data.

9 No. of 10. Which of the following is NOT a true statement? (A) Simple random sampling assures each element in population has equal probability of being chosen. (B) It is often difficult to measure an entire population. (C) Samples are more accurate than measuring an entire population. (D) A pre-election opinion poll is usually conducted using stratifies random sampling. (E) Systematic and Cluster are two other forms of sampling. This is a true statement, because simple random sampling assures subject independence. It is often difficult to measure an entire population, this is true and so we don t choose it. (C) Correct! Samples are more accurate that measuring an entire population. A pre-election opinion poll is usually conducted using stratifies random sampling, this is correct, so we do not choose it. This is a true statement. They are forms of sampling. (A). Incorrect! This is a true statement, because simple random sampling assures subject independence. (B). Incorrect! It is often difficult to measure an entire population, this is true and so we don t choose it. (C). Correct! Samples are more accurate than measuring an entire population, this is not true, and so we choose it. (D). Incorrect! A pre-election opinion poll is usually conducted using stratifies random sampling. This is a true statement. They are forms of sampling. (C) Samples are more accurate than measuring an entire population.

10 No. 10 of A company is interested in testing the effect of blood pressure control drug. So they have participants in its program using the new drug. Suppose we know that population mean is 10 and σ is also 10, then which of the following is NOT correct? (A) For a sample of 25 people, standard deviation=10/ 25=2. (B) For a sample of 36 people, standard deviation=10/ 36=1.67. (C) For a sample of 25 people, 68% chance sample mean is between 8 and 12. (D) For a sample of 4 people, standard deviation = or 10=7/10. (E) For a sample of 25 people, 5% chance sample mean is between 6 and 12. For a sample of 25 people, standard deviation=10/ 25=2, this is correct. For a sample of 36 people, standard deviation=10/ 36=1.67, this is correct. For a sample of 25 people, 68% chance sample mean is between 8 and 12 because with 68% probability it will fall between one standard deviation. This is not the formula for standard deviation. It should be 10/s4 = 10/7. This is the correct choice.. For a sample of 25 people, 5% chance sample mean is between 6 and 12. If numerous samples of the same size are taken, the frequency curve of means from various samples will be approximately normal. Mean will be same as mean for the population. If we let σ= population standard deviation and n = sample size, Standard deviation =σ/ n It should be 10/ 4 = 10/7. (D) For a sample of 4 people, standard deviation = or 10=7/10.