MTH – 245 (Statistics I) Fall 2021 Midterm.
Q1. (a)
Police plan to enforce speed limits during the morning rush hour on four different routes into the city. The traps on route A, B, C and D are operated 40%, 30%, 20% and 30% of the time respectively. Jack always speeds to work, and he has probability 0.2, 0.1, 0.5 and 0.2 of using those routes.
(i). What is the probability that he will get a ticket on any one morning?
Soln.
(ii). What is the probability that Jack will go five mornings without a ticket?
Soln.
Q1. (b)
In an Urn are 5 blue, 3 red and 2 yellow marbles. If you draw 3 marbles, what is the probability that less than 2 will be red if
(i). You draw with replacement
Soln.
(ii). You draw without replacement
Soln.
Q2. (a)
Determine whether or not the table is a valid probability distribution of a discrete random variable. Explain fully
(i).
x 0 1 2 3 4
P(x) -0.25 0.50 0.35 0.10 0.30
Soln.
(ii).
x 1 2 3
P(x) 0.325 0.406 0.164
Soln.
(iii).
x 25 26 27 28 29
P(x) 0.13 0.27 0.28 0.18 0.14
Soln.
Q2. (b)
Seven thousand lottery tickets are sold for $5 each. One ticket will win $2,000, two tickets will win $750 each, and five tickets will win $100 each. Let X denote the net gain from the purchase of a randomly selected ticket.
(i). Construct the probability distribution of X
Soln.
(ii). Is this a valid discrete probability distribution?
Soln.
(iii). Compute the expected value E(X) of X and interpret it’s meaning.
Soln.
(iv). Compute the standard deviation σ of X and interpret it’s meaning.
Soln.
Q3. (a)
Most graduate school of business require applicants for admission to take the Graduate Management Admission Council’s GMAT examination. Scores on GMAT are roughly normally distributed with a mean of 527 and a standard deviation of 112.
(i). What is the probability of an individual scoring below 500 on the GMAT?
Soln.
(ii). What is the probability of an individual scoring above 500 on the GMAT?
Soln.
(iii). How high must an individual score on the GMAT to score to be in the 85th percentile?
Soln.
(iv). How high must an individual score on the GMAT in order to score in the highest 5%?
Soln.
Q3. (b)
The length of human pregnancies from conception to birth approximate a normal distribution with a mean of 266 days and a standard deviation of 16 days.
(i). What proportion of all pregnancies will last less than 240 days?.
Soln.
(ii). What proportion of all pregnancies will last between 240 and 270 days (Roughly between 8 and 9 months)?
Soln.
(iii). What length of time marks the top 10% of all pregnancies?
Soln.
(iv). What length of time marks the shortest 70% of all pregnancies?
Soln.
Q4. (a)
(i). Compute the mean and standard deviation of the sampling distribution of the sample mean when you plan to take an SRS of size 64 from a population with a mean of 44 and standard deviation of 16.
Soln.
(ii). In (i) above, repeat the calculations for a sample size of 576. Explain the effect of the sample size increase on the mean and standard deviation of the sampling distribution.
Soln.
(iii). You take an SRS of size 64 from a population with mean 82 and standard deviation of 24. According to the Central Limit Theorem, what is the approximate sampling distribution of the sample mean? Use the 95 parts of the 68-95-99.7 rule to describe the variability of Xbar.
Soln.
(iv). In the settings of (iii) above, suppose we increase the sample size to 2304. Use the 95 parts of the 68-95-99.7 rule to describe the variability of Xbar. Compare your results with those you found in (iii)
Soln.
Q4. (b)
(i). Incomes in a certain town are approximately normally distributed with a mean of $36,000 and a standard deviation of $7,000. A random sample of 75 households is taken. What is the probability that the sample mean is greater than $37,000?
[ Hint: find the mean and standard deviation of the sample mean first and use the z-score formula to find your probability]
Soln.
(ii). The contents of bottles of beer are normally distributed with a mean of 300 and a standard deviation of 5ml. if a six (6) pack is randomly selected, what is the probability that its total content exceeds 1776 ml?
[ Hint: First, find the sample mean Xbar., then find the mean and standard deviation of the sample mean and use the z-score formula to find your probability]
Soln.
MTH – 245 (Statistics I) Fall 2021 Midterm.
Q1. (a)
Police plan to enforce speed limits during the morning rush hour on four different routes into the city. The traps on route A, B, C and D are operated 40%, 30%, 20% and 30% of the time respectively. Jack always speeds to work, and he has probability 0.2, 0.1, 0.5 and 0.2 of using those routes.
(i). What is the probability that he will get a ticket on any one morning?
Soln.
(ii). What is the probability that Jack will go five mornings without a ticket?
Soln.
Q1. (b)
In an Urn are 5 blue, 3 red and 2 yellow marbles. If you draw 3 marbles, what is the probability that less than 2 will be red if
(i). You draw with replacement
Soln.
(ii). You draw without replacement
Soln.
Q2. (a)
Determine whether or not the table is a valid probability distribution of a discrete random variable. Explain fully
(i).
x 0 1 2 3 4
P(x) -0.25 0.50 0.35 0.10 0.30
Soln.
(ii).
x 1 2 3
P(x) 0.325 0.406 0.164
Soln.
(iii).
x 25 26 27 28 29
P(x) 0.13 0.27 0.28 0.18 0.14
Soln.
Q2. (b)
Seven thousand lottery tickets are sold for $5 each. One ticket will win $2,000, two tickets will win $750 each, and five tickets will win $100 each. Let X denote the net gain from the purchase of a randomly selected ticket.
(i). Construct the probability distribution of X
Soln.
(ii). Is this a valid discrete probability distribution?
Soln.
(iii). Compute the expected value E(X) of X and interpret it’s meaning.
Soln.
(iv). Compute the standard deviation σ of X and interpret it’s meaning.
Soln.
Q3. (a)
Most graduate school of business require applicants for admission to take the Graduate Management Admission Council’s GMAT examination. Scores on GMAT are roughly normally distributed with a mean of 527 and a standard deviation of 112.
(i). What is the probability of an individual scoring below 500 on the GMAT?
Soln.
(ii). What is the probability of an individual scoring above 500 on the GMAT?
Soln.
(iii). How high must an individual score on the GMAT to score to be in the 85th percentile?
Soln.
(iv). How high must an individual score on the GMAT in order to score in the highest 5%?
Soln.
Q3. (b)
The length of human pregnancies from conception to birth approximate a normal distribution with a mean of 266 days and a standard deviation of 16 days.
(i). What proportion of all pregnancies will last less than 240 days?.
Soln.
(ii). What proportion of all pregnancies will last between 240 and 270 days (Roughly between 8 and 9 months)?
Soln.
(iii). What length of time marks the top 10% of all pregnancies?
Soln.
(iv). What length of time marks the shortest 70% of all pregnancies?
Soln.
Q4. (a)
(i). Compute the mean and standard deviation of the sampling distribution of the sample mean when you plan to take an SRS of size 64 from a population with a mean of 44 and standard deviation of 16.
Soln.
(ii). In (i) above, repeat the calculations for a sample size of 576. Explain the effect of the sample size increase on the mean and standard deviation of the sampling distribution.
Soln.
(iii). You take an SRS of size 64 from a population with mean 82 and standard deviation of 24. According to the Central Limit Theorem, what is the approximate sampling distribution of the sample mean? Use the 95 parts of the 68-95-99.7 rule to describe the variability of Xbar.
Soln.
(iv). In the settings of (iii) above, suppose we increase the sample size to 2304. Use the 95 parts of the 68-95-99.7 rule to describe the variability of Xbar. Compare your results with those you found in (iii)
Soln.
Q4. (b)
(i). Incomes in a certain town are approximately normally distributed with a mean of $36,000 and a standard deviation of $7,000. A random sample of 75 households is taken. What is the probability that the sample mean is greater than $37,000?
[ Hint: find the mean and standard deviation of the sample mean first and use the z-score formula to find your probability]
Soln.
(ii). The contents of bottles of beer are normally distributed with a mean of 300 and a standard deviation of 5ml. if a six (6) pack is randomly selected, what is the probability that its total content exceeds 1776 ml?
