# Mathematics

**NAME______________________ Score ______ / 50**

**STAT 200: Introduction to Statistics**

**Homework #3**

**Clearly Indicate Your Final Answer**

**Total Points Earned:**

1. **______ / 3**

2. **______ / 3**

3. **______ / 7**

4. **______ / 7**

5. **______ / 3**

6. **______ / 4**

7. **______ / 5**

8. **______ / 5**

9. **______ / 5**

10. **______ / 8**

**Score ______ / 50**

1. (3 points) Determine whether the given procedure results in a binomial distribution (or a distribution that can be treated as binomial). For those that are NOT binomial, identify at least one requirement that is not satisfied.

a. (1 point) The YSORT method of gender selection, developed by the Genetics & IVF Institute, is designed to increase the likelihood that a baby will be a boy. When 291 couples use the YSORT method and give birth to 291 babies, the genders of the babies are recorded.

b. (1 point) In an Idaho Potato Company Commission survey of 1000 adults, subjects are asked to select their favorite vegetables, and responses of potatoes, corn, broccoli, tomatoes, or “other” were recorded.

c. (1 point) Ten different US Senators are randomly selected without replacement, and the numbers of terms they have served are recorded.

2. (3 points) In the New York State Win 4 lottery, you place a bet by selecting four digits. Repetition is allowed, and winning requires that your sequence of four digits matches the four digits that are later drawn. Assume that you place one bet with a sequence of four digits.

a. (1 point) Use the multiplication rule to find the probability that your first two digits match those drawn and your last two digits do not match those drawn. That is, find *P*(*MMXX*), where “*M*” denotes a match and “*X*” denotes a digit that does not match the winning number.

b. (1 point) Beginning with MMXX, make a complete list of the different possible arrangements of two matching digits and two digits that do not match, then find the probability for each entry in the list.

c. (1 point) Based on the results from part (b), what is the probability of getting exactly two matching digits when you select four digits for the Win 4 lottery game?

3. (7 points) Based on the American Chemical Society, there is a 0.9 probability that in the United States, a randomly select $100 is tainted with traces of cocaine. Assume that eight $100 bills are randomly selected.

a. (1 point) Find the probability that all of the $100 bills have traces of cocaine.

b. (1 point) Find the probability that exactly seven of the $100 bills have traces of cocaine.

c. (1 point) Find the probability that the number $100 bills with traces of cocaine is six or more.

d. (1 point) If we randomly select eight dollar bills, what is the expected number (i.e. mean) of bills that would have traces of cocaine?

e. (1 point) If we randomly select eight dollar bills, what is the standard deviation of the number of bills that would have traces of cocaine?

f. (1 point) Using the range rule of thumb, find the minimum usual value and maximum usual value for the number of bills with traces of cocaine if we randomly select eight dollar bills.

g. (1 point) If we randomly select eight dollar bills, is six an unusually low number for those with traces of cocaine?

4. (7 points) When testing a method of gender selection prior to childbirth, we assume that the rate of female births is 50%, and we reject that assumption if we get results that are unusual in the sense that they are very unlikely to occur with a normal 50% birth rate. In a preliminary test of the XSORT method of gender selection during in-vitro fertilization, 15 births resulted in 14 girls.

a. (1 point) Assuming a 50% rate of female births, find the probability that in 15 births, the number of girls is 14.

b. (1 point) Assuming a 50% rate of female births, find the probability that in 15 births, the number of girls is 15.

c. (1 point) Assuming a 50% rate of female births, find the probability that in 15 births, the number of girls is 13 or fewer.

d. (1 point) Assuming a 50% rate of female births, find the expected number (i.e. mean) of girls in 15 births.

e. (1 point) Assuming a 50% rate of female births, find the standard deviation of the number of girls in 15 births.

f. (1 point) Using the range rule of thumb, find the minimum usual value and maximum usual value for the number of girls in 15 births.

g. (1 point) Do these preliminary results suggest that the XSORT method is effective in increasing the likelihood of a baby being a girl? Why or why not?

5. (3 points) Assume that the waiting time to board the Metrolink train follows a uniform distribution between 0 and 12 minutes (that is, a train comes every 12 minutes, and you could arrive any time in that interval). If you have not seen the Metrolink schedule (so, you have no idea when it is coming), find the probability that:

a. (1 point) You have to wait more than 6 minutes.

b. (1 point) You wait between 3 and 8 minutes.

c. (1 point) You wait exactly 4 minutes.

6. (4 points) For the following standard normal distribution graphs (that is, *μ* = 0, *σ* = 1), find the area associated with the z-score or the z-score associated with the area.

a. (1 point) b. (1 point)

*Area = ?*

*Area = ?*

*z = -1.04*

*z = -1.04*

*z = -1.04*

*z = -1.07*

*z = 0.67*

c. (1 point) d. (1 point)

*0.3050*

*0.2061*

*z = ?*

*z = ?*

7. (5 points) For the following questions, assume that a randomly selected subject is given a bone density test. The test scores are normally distributed with a mean of 0 (*μ* = 0) and a standard deviation of 1 (*σ* = 1). In each problem, draw a graph and find the probability of the given z-score(s).

a. (1 point) Less than -0.19 (z < -0.19).

b. (1 point) Less than 1.96 (z < 1.96).

c. (1 point) Between 1.23 and 2.37 (1.23 < z < 2.37).

d. (1 point) Between -0.62 and 1.78 (-0.62 < z < 1.78).

e. (1 point) Greater than -3.90 (-3.90 < z).

8. (5 points) Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test).

a. (1 point) For a randomly selected adult, find the probability of an IQ greater than 70.

b. (1 point) Find the probability that a randomly selected adult has an IQ between 90 and 110 (referred to as the *normal* range).

c. (1 point) Find the probability that a randomly selected adult has an IQ between 110 and 120 (referred to as *bright normal* range).

d. (1 point) Find the first quartile, Q1, which is the IQ score separating the bottom 25% from the top 75%.

e. (1 point) Mensa International calls itself “the international high IQ society,” and it has more than 100,000 members. Mensa states that “candidates for membership of Mensa must achieve a score at or above the 98th percentile on a standard test of intelligence (a score that is greater than that achieved by 98 percent of the general population taking the test).” Find the 98th percentile for the population of the Wechsler IQ scores. This is the lowest score meeting the requirement for Mensa membership.

9. (4 points) When a water taxi sank in Baltimore’s Inner Harbor, an investigation revealed that the safe passenger load for the water taxi was 3500 pounds. It was also noted that the mean weight of a passenger was assumed to be 140 pounds. Assume a “worst-case” scenario in which all of the passengers are adult men. (This is a valid assumption since Baltimore frequently hosts conventions and conferences in which people of the same gender often travel in groups). Assume that weights of men are normally distributed with a mean of 182.9 pounds and a standard deviation of 40.8 pounds.

a. (1 point) If one man is randomly selected, find the probability that he weighs less than 174 pounds (the new value suggested by the National Transportation Safety Board).

b. (1 point) With a maximum load limit of 3500 pounds, how many male passengers are allowed if we assume a mean weight of 140 pounds?

c. (1 point) With a maximum load limit of 3500 pounds, how many male passengers are allowed if we assume a mean weight of 182.9 pounds.

d. (1 point) Why is it necessary to periodically review and revise the number of passengers that are allowed on board?

10. (8 points) The ages of the four presidents who were assassinated in office are {56, 49, 58, 46} for Lincoln, Garfield, McKinley, and Kennedy (respectively). Use these ages as the “population for assassinated presidents” and select a random sample (with replacement) of size 2.

a. (2 point) List the 16 possible samples.

b. (1 point) Find the mean of each sample, then construct a table representing the sampling distribution of the sample mean. In the table, combine values of the sample mean that are the same.

c. (1 point) Compare the mean of the population {56, 49, 58, 46} to the mean of the sampling distribution of the mean.

d. (1 point) Do the sample means target the value of the population mean? In general, do sample means make good estimators of population means? Why or why not?

e. (1 point) Find the variance of each sample, then construct a table representing the sampling distribution of the sample variance. In the table, combine values of the sample variances that are the same.

f. (1 point) Compare the variance of the population {56, 49, 58, 46} to the mean of the sampling distribution of the variance.

g. (1 point) Do the sample variances target the value of the population variance? In general, do sample variances make good estimators of population variances? Why or why not?

Page 5 of 5