Have you ever tried to get out of jury duty? About 25% of those called will find an excuse (work, poor health, travel out of town, etc.) to avoid jury duty.†

(a) If 11 people are called for jury duty, what is the probability that all 11 will be available to serve on the jury? (Round your answer to three decimal places.)


(b) If 11 people are called for jury duty, what is the probability that 5 or more will not be available to serve on the jury? (Round your answer to three decimal places.)


(c) Find the expected number of those available to serve on the jury. What is the standard deviation? (Round your answers to two decimal places.)

(d) How many people n must the jury commissioner contact to be 95.9% sure of finding at least 12 people who are available to serve? (Enter your answer as a whole number.)
people

For patients who have been given a diabetes test, blood-glucose readings are approximately normally distributed with mean 128 mg/dl and a standard deviation 8 mg/dl. Suppose that a sample of 3 patients will be selected and the sample mean blood-glucose level will be computed. Enter answers rounded to three decimal places.

According to the empirical rule, in 95 percent of samples the SAMPLE MEAN blood-glucose level will be between the lower-bound of _________ and the upper-bound of ______

Part 1 Binomial Distribution [Mark 20%/cancer type, 40% total mark]

Five year survival chance from any cancer depends on many factors like availability of treatment options, expertise of attending medical team and more. Five year survival rate is also an important measure and it is used by medical practitioners to report prognosis to patients and family. We will be analyzing five year survival rate of two types of cancer, very aggressive and very treatable cancer and to have comparative analysis of cancer in Norway.

1. Five year survival rate of breast cancer is 87.7%
2. Five year survival rate of esophageal cancer is 16.5%

(NOTE: due to limitation imposed by our available probability distribution table assume survival rate for breast cancer is 90% and for esophageal cancer is 20%)

To simplify our comparative analysis, we will assume 480 patients were admitted in January 2018. For each type of cancer:

1. Calculate the number of patients that are expected to survive at least 5 years, calculate variance and standard deviation expected for 5 year survival.
2. For a selected 15 patients in each category, calculate probability of at least 7 patients surviving past 5 years, compare your calculated value with estimated number from table in page 329. How accurate is our estimation? Do you prefer estimation or calculation methods?
3. Use the table on page 329 of your textbook to determine survival probability for 0 to 15 patients and complete table below.

4. Draw frequency distribution bar graph for each type of cancer
5. What is the minimum/maximum number of survivors expected for given cancer type at 5 years. [HINT: use mean and 3 standard deviations to report outcome].

Part 2 Normal distribution [Mark 30%]

Daily discharge from phosphate mine is normally distributed with a mean daily discharge of 38 mg/L and a standard deviation of 12 mg/L. What proportion of days will the daily discharge exceed 58 mg/L?

Part 3 Normal approximation of binomial Probability Distribution [Mark 30%]

Airlines and hotels often grant reservation in excess to their available capacity, to minimize loss and maximize profitability due to no shows. Suppose that the records of Air Georgian shows that on average, 10% of their prospective passengers will not show up at departure gates. If Air Georgian sells 215 tickets  and their plane has capacity for 200 passengers.

1. Use binomial probability distribution to calculate, the mean of passengers showing up at the gate out of the 215 reservations made
2. Calculate the standard deviation.
3. Calculate standard z score for 200 passengers showing up at the gates.
4. Using normal probability distribution, determine probability of at least 200 passengers will show up!
5. Determine probability of more than 200 passengers showing up at the gate?
6. Determine probability of all 215 passengers showing up at the gate!

n 1998, the Nabisco Company launched a “1000 Chips Challenge” advertising campaign in which it was claimed that every 18-ounce bag of their Chips Ahoy cookies contains 1000 chips (on average). A curious statistics student purchased 8 randomly selected bags of cookies and counted the chocolate chips. The data is given below:

1200 1019 1214 1087 1214 900 1200 825

a) The student concluded that the data was not normally distributed and wanted to use a Wilcoxon Signed-Rank test to test the company’s claim. What assumption is needed in this case?

b) Assuming the assumption in part a. is met, at the 1% significance level, do the data provide sufficient evidence to conclude that the average number of chocolate chips in a bag of Chips Ahoy cookies differs from 1000? Carry out the Wilcoxon Signed-Rank Test by hand.

( PLEASE SHOW ALL YOUR WORK)

You will need your ticker code (company abbreviation) for stock prices for this question. Use your ticker code to obtain the closing prices for the following two time periods to obtain two data sets:

March 2, 2019 to March 16, 2019

Data set A

February 16, 2019 to February 28, 2019

Data set B

Take the closing prices from data set B and add 0.5 to each one of them. Treat data sets A and B as hypothetical sample level data on the weights of newborns whose parents smoke cigarettes (data set A), and those whose parents do not (data set B).

a) Conduct a hypothesis test to compare the variances between the two data sets.

b) Conduct a hypothesis to compare the means between the two data sets. Selecting the assumption of equal variance or unequal variance for the calculations should be based on the results of the previous test.

c) Calculate a 95% confidence interval for the difference between means

8) What proportion of a normal distribution is located between each of the following z-score boundaries?

a. z = –0.25 and z = +0.25

b. z = –0.67 and z = +0.67

c. z = –1.20 and z = +1.20

13) A normal distribution has a mean of μ = 30 and a standard deviation of σ = 12. For each of the following scores, indicate whether the body is to the right or left of the score and find the proportion of the distribution located in the body.

a. X = 33

b. X = 18

c. X = 24

d. X = 39

19) A report in 2010 indicates that Americans between the ages of 8 and 18 spend an average of μ = 7.5 hours per day using some sort of electronic device such as smart phones, computers, or tablets. Assume that the distribution of times is normal with a standard deviation of σ = 2.5 hours and find the following values.

a. What is the probability of selecting an individual who uses electronic devices more than 12 hours a day?

b. What proportion of 8- to 18-year-old Americans spend between 5 and 10 hours per day using electronic devices? In symbols, p (5 < X < 10) = ?

Please show Excel work:

A. Use α = .05. Test to determine whether the proportions of female and male voters who intend to vote for the Democrat candidate differ? Report the test statistic and the p-value.

B. Provide a 99% confidence interval for the difference in the proportion of female and male voters who intend to vote for the Democrat candidate

1. What demographic variables were measured at the nominal level of measurement in the Oh et al. (2014) study? Provide a rationale for your answer. 2. What statistics were calculated to describe body mass index (BMI) in this study? Were these appropriate? Provide a rationale for your answer. 3. Were the distributions of scores for BMI similar for the intervention and control groups? Provide a rationale for your answer. 4. Was there a signifi cant difference in BMI between the intervention and control groups? Provide a rationale for your answer.

We are interested in whether math score (math – a continuous variable) is a significant predictor of science score (science – a continuous variable) using the High School and Beyond (hsb2) data.

State the null and alternative hypotheses and the level of significance you intend to use.

Ho:β=0

H1:β≠0

Alph:0.05

Write the equation for the appropriate test statistic.

t =b/SE(b)

What is your decision rule? Be sure to include the degrees of freedom.

If our t value is greater than the critical value of 1.96 we reject the null hypothesis.

FD= n-2=200-2= 198=1.96

Using SAS, estimate the means, variances and covariances for math and science scores. Copy and paste the relevant SAS output below.

Using the output from (d), calculate by hand the slope. Be sure to show your work.

Using the output from (d), calculate by hand the intercept. Be sure to show your work

Baseball's World Series is a maximum of seven games, with the winner being the first team to win four games. Assume that the Atlanta Braves and the Minnesota Twins are playing in the World Series and that the first two games are to be played in Atlanta, the next three games at the Twins' ballpark, and the last two games, if necessary, back in Atlanta. Taking into account the projected starting pitchers for each game and the home field advantage, the probabilities of Atlanta winning each game are as follows:

a. Set up a spreadsheet simulation model for which whether Atlanta wins or loses each game is a random variable. What is the probability that the Atlanta Braves win the World Series? If required, round your answer to two decimal places.

b. What is the average number of games played regardless of winner? If required, round your answer to one decimal place.

Assume that a sample is used to estimate a population proportion p. Find the 95% confidence interval for a sample of size 380 with 125 successes. Enter your answer as a tri-linear inequality using decimals (not percents) accurate to three decimal places.

___ < p < ____

Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

The director of research and development is testing a new medicine. She wants to know if there is evidence at the 0.1 level that the medicine relieves pain in more than 363 seconds. For a sample of 57 patients, the mean time in which the medicine relieved pain was 367 seconds. Assume the population standard deviation is 24. Find the P-value of the test statistic.

Find the median, the lower half and the upper half of the history 108 test scores 10,16,14,22,21,13,15,14,10,18,19,8,16,12,18,11,9,10,15,10,21,14,18,19,1819,3,25,18,13,1,16,9,14,821,13,14,18,16,5,11,17,14,12,16,18,16,18,17,10,12,19,9,3,15,17

Time spent using e-mail per session is normally distributed,
with m = 9 minutes and s = 2 minutes. If you select a random
sample of 25 sessions,
a. what is the probability that the sample mean is between 8.8 and
9.2 minutes?
b. what is the probability that the sample mean is between 8.5 and
9 minutes?

c. If you select a random sample of 100 sessions, what is the prob-
ability that the sample mean is between 8.8 and 9.2 minutes?

d. Explain the difference in the results of (a) and (c).

When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ.

Method 1: Use the Student's t distribution with d.f. = n − 1.
This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method.

Method 2: When n ≥ 30, use the sample standard deviation s as an estimate for σ, and then use the standard normal distribution.
This method is based on the fact that for large samples, s is a fairly good approximation for σ. Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution.

Consider a random sample of size n = 31, with sample mean x = 44.4 and sample standard deviation s = 4.7.

(a) Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

(b) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

(c) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

No. The respective intervals based on the t distribution are shorter.Yes. The respective intervals based on the t distribution are shorter.     Yes. The respective intervals based on the t distribution are longer.No. The respective intervals based on the t distribution are longer.

(d) Now consider a sample size of 71. Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

(e) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

(f) Compare intervals for the two methods. Would you say that confidence intervals using a Student's t distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution?

No. The respective intervals based on the t distribution are shorter.No. The respective intervals based on the t distribution are longer.     Yes. The respective intervals based on the t distribution are longer.Yes. The respective intervals based on the t distribution are shorter.

With increased sample size, do the two methods give respective confidence intervals that are more similar?

As the sample size increases, the difference between the two methods becomes greater.As the sample size increases, the difference between the two methods remains constant.     As the sample size increases, the difference between the two methods is less pronounced.