5. Decide whether each of the statements below is true or false.

(a) The mean of the first test scores is 81. This means that 50% of the class got a score less than 81.

(b) A distribution is skewed to the left. The mean is less than the median.

(c) The mean does not have to be one of the observations in a data set, while the median has to be one of the observations in the data set.

(d) The mean is resistant to outliers; the median is not.

(e) The sum of the deviations from the mean reflects the amount of variability in a data set.

(f) The standard deviation of a variable (i.e., not a constant) is always positive, and it cannot be zero or negative.

(g) If all the observations have the same value, the range, the variance, the standard deviation and the IQR are all equal to zero.

(h) The range, the standard deviation, the IQR (but not the variance) are all expressed in the same units as the measurements used in the data set.

(i) Adding a constant number to each observation in the data set does not change the range, variance, standard deviation and IQR.

(j) If the weights of females measured in pounds are converted to kilograms, the numeric value of the sample standard deviation does not change.

6. In a nonrandomized clinical trial on patients with disease X with no known treatment, two groups are being compared. Group A consists of individuals who take drug A in addition to standard of care; group B consists of individuals who receive standard of care. Table 1 reveals that there are noteworthy differences between the two groups at baseline. The variables listed in Table 1 (such as history of hypertension and diabetes status) suggest that patients in Group B are not as unhealthy as those in Group A. As it turns out, at the end of the study, those who are on Drug A generally have a better survival outcome than those who receive only standard of care. A researcher suggests that an RCT is not necessary to show conclusively the efficacy of Drug A because Drug A has given the group of individuals with comorbidities a survival advantage. Do you agree with this statement? Explain your answer briefly.

7. One of the variables measured in a study was height in, which was height of an individual at baseline measured in inches.

(a) Suppose a new variable height cm was created, which was height in converted to cen timeters. The sample mean was going to change in value. For example, if the mean height was 65 inches, the new mean height would be 165 cm. Note that 65 inches and 165 cm, albeit different numbers, are really the same quantity. In addition to the mean, other statis tics will also change in value. Which of the following statistics is unaffected by the change in the unit of measurement? range, IQR, standard deviation, variance, coefficient of varia tion, minimum, maximum, Q1, Q2

(b) Suppose it was found out that the weighing scale always underestimated a person's weight by 6 lbs. It was thus necessary to add 6 lbs to weight. A new variable weight plus6 was created. The new mean weight is going to increase by 6 lbs. In addition to the mean, other statistics will also change in value. Which of the following statistics is unaffected by the addition of 6 lbs? range, IQR, standard deviation, variance, coefficient of variation, minimum, maximum, Q1, Q2

8. (a) The mean homework score of 20 students is 82. Is it possible to find the sum of the 20 homework scores? If yes, what is it?

(b) The median homework score of 20 students is 82. Is it possible to find the sum of the 20 homework scores? If yes, what is it?

9. Study the table below which I took from the CDC on incidence of prostate cancer in different age groups among male seniors (70 years old or older) in 2014. The first two columns provide an age distribution for male seniors who did not have prostate cancer before 2014; the third column gives the percentage of new cases of prostate cancer in 2014 in each age group.

Age Group % Seniors % Prostate

(year) Cancer

70-74 39.0% 0.52%

75-79 30.4% 0.44%

80-84 18.3% 0.34%

85+ 12.3% 0.31%

100.0%

The incidence of prostate cancer decreases as age increases. What must we bear in mind when interpreting this statement?

10. In an open access data set on 828 COVID-19 patients worldwide, 59% were male and 8% had diabetes. Suppose 3% of the COVID-19 patients were diabetic males. A person is selected at random in this sample.

(a) What is the probability that the COVID-19 patient is diabetic or male?

(b) What is the probability that the COVID-19 patient is a diabetic female? (c) What is the probability that the COVID-19 patient is a nondiabetic male?

(d) What is the probability that the COVID-19 patient in this sample is a nondiabetic female?

(e) If the COVID-19 patient is diabetic, what is the probability that this person is male?

(f) What percentage of diabetic COVID-19 patients in this sample is male?

(g) What percentage of nondiabetic COVID-19 patients in this sample is male?

(h) Of the nondiabetic COVID-19 patients, what percentage is female?

Show your work to receive full credit. (You may use a rule of probability or draw a Venn diagram.)