The Body Mass Index (BMI) is a value calculated based on the weight and the height of an individual. In a small European city, a survey was conducted one year ago to review the BMI of the citizens. In the sample

22
with 200 citizens, the mean BMI was 23.3 kg/m and standard deviation was 1.5 kg/m . It is reasonable to

assume the BMI distribution is a normal distribution.

1. (a) Find the point estimate of the population mean BMI one year ago.

2. (b) Calculate the sampling error at 90% confidence.

3. (c) Construct a 90% confidence interval estimate of the population mean BMI one year ago.

This city launched a healthy exercise program to reduce citizen’s BMI after last year’s survey.
program effectively reduces the BMI of each citizen by 2.5%.
(d) Construct a 98% confidence interval estimate of the population mean BMI after the healthy exercise

program. (Hint: find the updated sample mean and sample standard deviation of the BMI of the sample with 200 citizens selected last year)

Let b > 0 be an integer. Find the probability that a symmetric simple random walk started from 0 visits b the first time in the nth step.

Hint: Draw a picture, and try to describe the requirements that the path consisting the first n−1 steps should satisfy. The Reflection principle (or a related result) should be helpful after that.

A random sample of

91

observations produced a mean

x=26.2

and a standard deviation

s=2.6

a. Find a​ 95% confidence interval for

μ.

b. Find a​ 90% confidence interval for

μ.

c. Find a​ 99% confidence interval for

μ.

The lengths of pregnancies are normally distributed with a mean of 272 days and a standard deviation of 16 days.

1. Find the probability that a pregnancy lasts more than 295 days.
2. If we stipulate that a baby is premature if the length of pregnancy is in the lowest 3%, find the length of pregnancy that separates premature babies from those who are not premature.

1) Use either the critical value or p-value method for testing hypotheses.
2) Identify the null and alternative hypotheses, test statistic, P-value (or range of P-values), and critical value(s).
3) State your final conclusion that addresses the original claim. Include a confidence interval as well and restate this in your original conclusion.

In a random sample of 300 patients, 21 experienced nausea. A drug manufacturer claims that fewer than 10% of patients who take its new drug for treating Alzheimer’s disease will experience nausea. Test this claim at the 0.05 Significance Level.

A study is conducted to assess whether residents in City A spent a different out-of-pocket amount on prescription medications from residents in City B last year. The study is restricted to residents who are 50 years of age or older. Residents are selected at random. For each resident, the total amount of dollars spent on prescription medications over the last year is recorded. The summary statistics of the sample data are given in the table below.

Let μ1 be the mean out-of-pocket amount that residents in City A spent, and μ2 be the mean out-of-pocket amount that residents in City B spent. Run a two-sample t-test assuming equal variances. Use a significance level of 0.05.

City Sample size Sample mean Sample standard deviation

A 40 381 39 B 52 422 45

1. Write down the null hypothesis. (5 points)

2. Write down the alternative hypothesis. (5 points)

3. Calculate the point estimate for μ1 − μ2. (5 points)

4. Calculate the pooled sample variance. (10 points)

5. Calculate the standard error of the point estimate in c. (10 points)

6. Calculate the test statistic. (10 points)

7. Find out the critical value. (5 points)

8. Is there a statistically significant difference in the out-of-pocket amount spent on prescription medications between City A and City B? (5 points)

A generic brand of all-purpose fertilizer G claims to contain 10 percent phosphorous, a nutrient which helps plants grow strong root systems. A leading name brand of all-purpose fertilizer N advertises having a phosphorous concentration at least 30 percent higher than the generic brand. The most important consideration is that the mean phosphorous concentration in each brand is at least as high as advertised; a slightly higher phosphorous concentration is not concerning, but having too little phosphorous will hurt or slow the growth of plants’ root systems. Further, brand N is significantly more expensive than brand G per pound, so consumers who purchase brand N fertilizer expect the product to produce stronger root systems as advertised.

You work for a consumer protection group that has received complaints about both brands. The complainants claim that the mean phosphorous concentration in each fertilizer is lower than advertised. Additionally, you wish to determine whether brand N’s claim of having a phosphorous concentration at least 30% higher than brand N is true. If any of the companies’ claims are found to be untrue, the consumer protection group will bring a class-action lawsuit against the brand or brands responsible.
The consumer protection group bought bags of each brand of fertilizer and sent them to a lab for analysis. The phosphorous content (as a percent) for each bag, along with a column indicating which brand produced the bag is detailed below:

Phosphorous (% per bag), Brand
9.79522878032876, G
10.9302191178489, G
12.1732852582726, N
9.0544640624081, G
10.0475170197499, N
11.6656993826563, N
12.6642653530907, N
9.10468690374451, G
9.03436435420168, G
13.7352205503801, N
14.5489736025146, N
9.77948772403382, N
12.2986303191169, N
10.3807530975497, G
12.1950434195649, G
9.36377060086138, G
9.35399703787208, G
11.6263999062187, N
14.8368893022518, N
10.3076684543705, G
8.95153287738328, G
9.38069973700502, G
12.2352561489847, N
8.75328753808385, G
7.67158718250212, G
11.060128497079, N
13.3621542490895, N
12.1468548485131, N
9.32726632730324, N
12.2110173028274, N
9.41454096948369, G
10.4809518447645, N
9.96123587641246, G
10.1635577800051, G
10.8408606925903, N
16.1557438655007, N
7.94990087762832, N
9.44666693523413, G
9.16255880009391, G
10.8115193560222, G

a) What is the appropriate hypothesis test to determine whether brand G has the mean phosphorous concentration advertised?
b) What is the appropriate hypothesis test to determine whether brand N has the mean phosphorous concentration advertised?
c) What is the appropriate hypothesis test to determine whether brand N has a mean phosphorous concentration at least 30% higher than brand G as advertised?

1. A corporation owns several companies. The strategic planner for the corporation believes amount spent on advertising can to some extent be a predictor of total sales. As an aid in long term planning, she gathers the following sales and advertising information from several of the companies in millions for the year 2019.

                Advertising          12.5        3.7          21.6        60.0        37.6        6.1          16.8        41.2

                Sales                    148        55           338         994         541         89           126         379

1. Develop the equation of the simple regression line to predict sales on advertising expenditures using these data.                                                                                                             
2. Find out the sales that are likely to be attained when advertising expenditure is 25

million.                                                                                                          

3. Compute the coefficient of correlation and coefficient of determination and interpret their values.
4. Test the hypothesis that the population correlation coefficient is zero at 5% level of significance.

1. Biodiesel fuel has a cloud point, the temperature at which the fuel becomes cloudy, of approximately 13 ⁰C . This clouding can lead to poor engine performance and can even cause an engine to stop completely. An industrial chemical company produces an additive designed to lower the cloud point of this type of fuel. A random sample of six different biodiesel fuels was obtained and the cloud point was measured for each. One gram of the chemical additive was mixed in with every fuel sample and the cloud point was measured again. The resulting data are

given in the following table ( temperature in ⁰C ).

Assume normality and conduct the appropriate hypothesis test to determine whether the additive lowers the mean cloud point in biodiesel fuel. Use α = 0.05.

Show your calculations in detail please..

1. Suppose that a researcher is conducting a study on fast food habits and assumes that 55% of adults prefer hamburger to pizza. He randomly selects 10 adults and ask whether they prefer hamburger to pizza or not. Based on this information, answer the following questions.

a) (20 pts) Define the random variable (in words) and check whether it is a binomial random variable or not and write your conclusion. Explain your answer.

b) (10 pts) What is the probability that at least 8 adults prefer hamburger?

c) (10 pts) What is the probability that as most 3 adults prefer hamburger?

d) (15 pts) What is the expected number of people who prefer hamburger.

e) (15 pts) Calculate the standard deviation of this random variable and interpret it.

1. Positive behaviors on the normal curve has a mean of 12 and a standard deviation of 3, what are the z-scores and percentiles for the following outcomes?

1. 15 positive behaviors
2. 18 positive behaviors
3. 21 positive behaviors
4. 9 positive behaviors
5. 6 positive behaviors
6. 3 positive behaviors
7. 12 positive behaviors

In order to compare the satisfaction of customers of 2 competitor companies of , 174 customers of company A and 355 customers of company B were chosen randomly. Customers were required to rate the companies from level 1 (least satisfaction) to 5 (maximum satisfaction). The results are included below. Check whether the difference in the mean level of customers' satisfaction of the two companies is statistical important at 1%. The results are included below:

Company A   Company B
Sample Size   174   355
Sample mean   3.51   3.24
Sample Standard Deviation (S)   0.51   0.52

There are 6 Blue and 6 Red chips in a box. We take 4 chips without replacement. The number of taken Blue chips is shown by X and the number of taken Red chips is shown by Y. the correlation coefficient between X and Y equals:

Describe some sampling situations in which a sampler which takes a stratified sample would be necessary.

The director of admissions at a large university advises parents of incoming students about the cost of textbooks during a typical semester. He selected a sample of 100 students and recorded their textbook expenses for the semester. He then calculated a sample mean of $675.60 and a sample standard deviation of $45.20. You may assume that the distribution of textbook expenses is approximately normally distributed.

(a) Is there sufficient evidence that the population mean textbook expense per semester is above $665? You should justify your answer with a hypothesis test at the 5% significance level.

(b) Construct a 95% confidence interval estimate of the population mean textbook expense per semester.

Please do not include any image.