Using R-studio

2. Consider an experiment where we flip a fair coin six times in a row, and i is the number of heads tossed:

            a.         Calculate the probability mass function for i = 0. . . 6 using the equation from Ross section 2.8 for Binomial Random Variables

            b.         Conduct a simulation of this experiment in R, with T trials of the experiment – pick several values of T from 10 to 10,000.

            c.         Create a plot of the theoretical result vs. your simulation at T = 100 and T = 10,000. Show that they converge as T increases.

We have a bag filled with 201 marbles, of which 100 of them are blue and 101 of them are red. Every turn, we remove 2 marbles from the bag. If the two marbles are of the same color, we remove the two marbles but add a blue marble into the bag. If the two marbles are of different colors, we remove the two marbles and add a red marble into the bag. What is the color of the last marble in the bag?

Below is a data table from a survey of 12 random car owners that were asked about their typical monthly expenses in dollars on gasoline.

210 160 43 255 176 135 221 359 380 405 391 477

1. (a) Is there evidence at the 5% significance level to suggest the mean monthly gasoline expense for all car owners is less than $200?

2. (b) Interpret the P-value in the context of this test.

3. (c) Explain what a Type I error would mean in the context of this test.

4. (d) Explain what a Type II error would mean in the context of this test.

Using the Newsboy Model

1.Needless Markup (NM), a famous “high end” department store, must decide on the quantity of a high-priced woman’s handbag to procure in Spain for the coming Christmas season. The unit cost of the handbag to the store is $28.50 and the handbag will sell for $150.00. Any handbags not sold by the end of the season are purchased by a liquidator for $10.00 each. In addition, the store accountants estimate that there is a cost of $0.40 for each dollar tied up in inventory, as this dollar invested elsewhere could have yielded a gross profit. Assume that this cost is attached to unsold bags only.

Answer the following questions:

1. Due to the long distance and limited capacity, NM must place the order 6 months in advance. A detailed analysis of past data shows that if forecasting 6 month in advance, the number of bags sold can be described by a normal distribution, with mean 150 and standard deviation 60. What is the optimal number of bags to purchase?
2. What is the expected cost of mismatch under the optimal purchase quantity? What is the optimal expected profit?
3. Another supplier in the U.S. offers the same product but at a higher price of $35 due to its higher production cost. For this supplier, NM only needs to place the order 3 months in advance which results in a much better forecast. Past data shows if ordering 3 months in advance, the number of bags sold can be described by a normal distribution, with mean 150 and standard deviation 20. Which supplier should NM choose?

1. The data in BUSI1013 Credit Card Balance.xlsx is collected for building a regression model to predict credit card balance of retail banking customers in a Canadian bank. Use this data to perform a simple regression analysis between Account balance and Income (in thousands). (12 points)
   
2. Develop a scatter diagram using Account Balance as the dependent variable y and Income as the independent variable x.
3. Develop the estimated regression equation.
4. Use the estimated regression equation to predict the Account Balance of a customer with Income of $58 thousands.
5. Use the critical-value approach to perform an F test for the significance of the linear relationship between account balance and Income at the 0.05 level of significance.
6. What percentage of the variability of Account Balance can be explained by its linear relationship with Income?
7. Use the p-value approach to perform a t test for the significance of the linear relationship between Account Balance and Income at the 0.05 level of significance.

The average number of words in a romance novel is 64,419 and the standard deviation is 17,160. Assume the distribution is normal. Let X be the number of words in a randomly selected romance novel. Round all answers to 4 decimal places where possible.

a. What is the distribution of X? X ~ N(,)

b. Find the proportion of all novels that are between 71,283 and 83,295 words.

c. The 85th percentile for novels is  words. (Round to the nearest word)

d. The middle 60% of romance novels have from  words to  words. (Round to the nearest word)

1. ABC Corporation has 350 employees. The distribution of number of sick days per employee per year for the population of 350 employees is not highly skewed and has a mean of 12 and a standard deviation of 4. Suppose a simple random sample of 25 employees' number of sick days is taken, what is the probability that the sample of 25 will have mean number of sick days between 10 and 14? Show your work.

This study investigates the hours per month spent on the Internet by U.S. residents, age 18 to 24. The sample results are: n= 75, x-bar=28.5, and s=23.1 . We are to construct a 95% confidence interval based on this information.

a) What is critical value of t (that is, t*) for this confidence interval?

b) the standard error of the mean (SEM) is:

c) Rounded to one decimal place, as x-bar was, the margin of error (MOE) for your confidence interval will be:

d) Which of the following conclusions is CORRECT?

We are 95% sure the mean hours spent per week on the Internet by U.S. residents, age 18 to 24, is between 23.5 and 34.1.

We are 95% sure the mean hours spent per month on the Internet by college students is between 23.2 and 33.8.

We are 95% sure the mean hours spent per month on the Internet by U.S. residents, age 18 to 24, is between 23.2 and 33.8.

We are 95% sure the mean hours spent per month on the Internet by U.S. residents, age 18 to 24, is between 23.5 and 34.1.

1. In a study of the effectiveness of certain exercises in weight reduction, a group of 16 persons engaged in these exercises for three months and showed the weights, in pounds, before exercises (X) and after exercises (Y):  

1. Perform a pooled two-sample t-test at 0.05 level of significance to see if the data provide sufficient evidence to show a weight reduction. What assumptions are required for this test to be valid?
2. Perform a paired t-test at 0.05 level of significance to see if the data provide sufficient evidence to show a weight reduction. What assumptions are required for this test to be valid?
3. Are the conclusions from (a) and (b) consistent, and why or why not?

1. Facebook penetration values (the percentage of a country’s population that are Facebook users) for 15 randomly selected countries are: 52.56, 33.09, 5.37, 19.41, 32.52, 41.69, 51.61, 30.12, 39.07, 30.62, 38.16, 49.35, 27.13, 53.45, 40.01.
   1. Give point estimates for the population mean Facebook penetration (µ), and the population standard deviation of Facebook penetration (_σ).
   2. Construct a 95% confidence interval estimation for µ. Interpret.
   3. Construct a 95% confidence interval estimation for _σ. Interpret.
   4. What assumptions do you need to make about the population so that the

confidence intervals given in (b) and (c) are valid?

The effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 20.7 for a sample of size 464 and sample standard deviation 21.4.

Estimate how much the drug will lower a typical patient's systolic blood pressure (using a 99% confidence level).

Enter your answer as a tri-linear inequality accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).

< μμ <

Answer should be obtained without any preliminary rounding.

d) Interpret the confidence interval in the words of the problem.

Consider an i.i.d. random sample of size 3 denoted by ?1,?2, ?3 from the same population, where the mean ? and variance ? 2 are unknown. Suppose that you have the following two different estimators for mean ?. (Remember: no work, no credit.) ?̂1 = 0.3?1 + 0.5?2 + 0.2?3 ?̂2 = 0.5?1 + 0.5?3 a. Is ?̂1 unbiased? b. Is ?̂2 unbiased? c. Which one is preferred, ?̂1 or ?̂2?

Normal body temperature? A random sample of 20 healthy adults was collected and their body temperatures were measured in degrees Fahrenheit. The data can be found in an excel file Body with variable name Temperature. Do these data give evidence that the true mean body temperature for healthy adults is not equal to the traditional 98.6 degrees Fahrenheit? Test an appropriate hypothesis at 5% level of significance. (50 points total)

Source: Moore, D., Notz, W. and Fligner, M. (2015). The Basic Practice of Statistics (7^th edition). New York, NY: W. H. Freeman and Company.

Label the parameter: (4 points)

State null and alternative hypotheses: (6 points)

H₀:

H_a:

Propose an appropriate hypothesis test. Explain! (4 points)

Verify the required conditions for the proposed hypothesis test: (8 points)

Randomization assumption:

Normality assumption:

Note: For your future reference, save your R codes and normal QQ plot on your machine:

Using R, report test statistic, df, and p-value. (9 points)

Test statistic =

df =

p-value =

         Note: For your future reference, save your R codes and outputs on your machine:

Interpret test statistic: (4 points)

Interpret the P-value in context: (6 points)

Make a decision of hypothesis test: (4 points)

Make a conclusion of hypothesis test in context: (5 points)

1. Based on historical data, your manager believes that 31% of the company's orders come from first-time customers. A random sample of 146 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is greater than than 0.22?
Note: You should carefully round any z-values you calculate to 4 decimal places to match wamap's approach and calculations.

2. Business Weekly conducted a survey of graduates from 30 top MBA programs. On the basis of the survey, assume the mean annual salary for graduates 10 years after graduation is 157000 dollars. Assume the standard deviation is 39000 dollars. Suppose you take a simple random sample of 84 graduates.
Find the probability that a single randomly selected salary is less than 160000 dollars.
Answer =
Find the probability that a sample of size n=84 n=84 is randomly selected with a mean that is less than 160000 dollars.
Answer =
Enter your answers as numbers accurate to 4 decimal places.

A corporation randomly selects 150 salespeople and finds that​ 66% who have never taken a​ self-improvement course would like such a course. The firm did a similar study 10 years ago in which​ 60% of a random sample of 160 salespeople wanted a​ self-improvement course. The groups are assumed to be independent random samples. Let

pi 1π1

and

pi 2π2

represent the true proportion of workers who would like to attend a​ self-improvement course in the recent study and the past​ study, respectively.

What is the critical value when performing a ​chi-square test on whether the population proportions are different if alpha a ​= 0.05?