There is a lot of interest in the relationship between studying music and studying math. We will look at some sample data that investigates this relationship. Below are the Math SAT scores from 8 students who studied music through high school and 11 students who did not. Test the claim that students who study music in high school have a higher average Math SAT score than those who do not. Test this claim at the 0.05 significance level.

The 8 students who studied music in high school (x₁)

The 11 students who did not study music in high school (x₂)

If you are using software, you should be able copy and paste the data.

(b) Use software to calculate the test statistic or use the formula

t =

(c) Use software to calculate the degrees of freedom (d.f.) or use the formula

Round your answer to the nearest whole number.
d.f. =   

(d) What is the critical value of t? Use the answer found in the t-table or round to 3 decimal places.
t_α =  

The following results are from an independent-measures, two factor study. Be careful that this is a 2x3 design. You are expected to extend your knowledge on 2x2 design to a 2x3 factorial design.

The study:

Consider an experiment designed to investigate the effectiveness of therapy for the treatment of anxiety or fear. Two kinds of therapy (systematic desensitization, counter conditioning) as well as a counseling-only control condition are included. The therapy and counseling programs have been conducted over a number of sessions and the investigator asks whether more sessions might bring further improvements. The goal then is to compare two treatment lengths: the original and an extended version. For convenience, these two lengths are referred to as “Short” and “Long.” Participants’ well-being scores are measured; the larger the score, the more beneficial the treatment. Data are given in the table below. Answer the following questions using this data.

1. What are the dependent and independent variables in this study?
2. How many levels does each independent variable have and what are they?
3. State the hypotheses for each test.
4. Use a two-factor ANOVA with a=.05 to evaluate the main effects and interaction. For your answer, create a summary table and fill in the values. Do not show calculations here.
5. Based on what you found in ‘d’, what is your decision for each hypothesis test? Give F_obsand F_critfor each hypothesis test.
6. Plot cell means (you can draw this by hand if you don’t know how to do it using a computer) with form of therapy on the x-axis, participants’scoreson the y-axis, and duration of therapy as the grouping variable.
7. Run post-hoc analysis to sort out the interaction between two independent variables (simple effects analysis).
8. Based on your decision on the main effects and interaction, interpret the results. The interaction should have come out significant, so interpret accordingly! Here, I am not asking you to write a conclusion as it would appear in a research paper. I am simply asking you to interpret the results in your own words. The answer to this question should not take more than 5-6 simple sentences. If you are writing more than that, you are being redundant!

We have the survey data on the body mass index (BMI) of 659 young women. The mean BMI in the sample was

x = 26.2. We treated these data as an SRS from a Normally distributed population with standard deviation σ = 8.1.

Give confidence intervals for the mean BMI and the margins of error for 90%, 95%, and 99% confidence. (Round your answers to two decimal places.)

confidence level interval margin of error

90% _____ to _____   _____

95%    _____ to _____   _____

99%        _____ to _____   _____

It is estimated that 27% of births at a particular hospital are by caesarian section. Suppose a sample of 5 births is taken from this hospital.

(a). What is the probability that at most 2 births will be by caesarian section in this hospital?



(b). Write the R code to compute the probability in part (a).



(c). What is the mean number of births delivered by caesarian section in this hospital?



(d). What is the standard deviation of the number births delivered by caesarian section in this hospital?

A certain mutual fund invests in both U.S. and foreign markets. Let x be a random variable that represents the monthly percentage return for the fund. Assume x has mean μ = 1.8% and standard deviation σ = 0.7%.

(a) The fund has over 250 stocks that combine together to give the overall monthly percentage return x. We can consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all world stocks. Then we see that the overall monthly return x for the fund is itself an average return computed using all 250 stocks in the fund. Why would this indicate that x has an approximately normal distribution? Explain. Hint: See the discussion after Theorem 6.2.

The random variable  x / x-bar is a mean of a sample size n = 250. By the theory of normality / law of large numbers / central limit theorem, the x-bar / x distribution is approximately normal.

(b) After 6 months, what is the probability that the average monthly percentage return x will be between 1% and 2%? Hint: See Theorem 6.1, and assume that x has a normal distribution as based on part (a). (Round your answer to four decimal places.)


(c) After 2 years, what is the probability that x will be between 1% and 2%? (Round your answer to four decimal places.)


(d) Compare your answers to parts (b) and (c). Did the probability increase as n (number of months) increased?

Yes OR No    

Why would this happen?

The standard deviation Increases / Decreases / Stays the same as the Sample size / Average / Mean / Distribution increases.

(e) If after 2 years the average monthly percentage return was less than 1%, would that tend to shake your confidence in the statement that μ = 1.8%? Might you suspect that μ has slipped below 1.8%? Explain.

This is very unlikely if μ = 1.8%. One would suspect that μ has slipped below 1.8%

This is very likely if μ = 1.8%. One would suspect that μ has slipped below 1.8%

This is very likely if μ = 1.8%. One would not suspect that μ has slipped below 1.8%

This is very unlikely if μ = 1.8%. One would not suspect that μ has slipped below 1.8%.

You get bit by a zombie. You have 3 antidote pills but they have been mixed in a bottle with 4 aspirin and 5 heart pills. If you take more than 4 heart pills, then you will die. Therefore, you choose 4 pills at random. What is your probability of survival? That is, what is the probability that you randomly choose at least one antidote pill?

In a recent study on world​ happiness, participants were asked to evaluate their current lives on a scale from 0 to​ 10, where 0 represents the worst possible life and 10 represents the best possible life. The mean response was 5.2 with a standard deviation of 2.6. ​(a) What response represents the 94th ​percentile? ​(b) What response represents the 61st ​percentile? ​(c) What response represents the first ​quartile?

A poll of 809 adults aged 18 or older asked about purchases that they intended to make for the upcoming holiday season. One of the questions asked about what kind of gift they intended to buy for the person on whom they would spend the most. Clothing was the first choice of 498 people. Give a 99% confidence interval for the proportion of people in this population who intend to buy clothing as their first choice.

PLEASE WALK ME THROUGH THE STEPS! THANK YOU!

10. A manufacturer produces 25-pound lifting weights. The lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely so the distribution of weights is uniform. A sample of 100 weights is taken.

a. Find the probability that the mean actual weight for the 100 weights is greater than 24.8. (Round your answer to four decimal places.

b. Find the 95th percentile for the mean weight for the 100 weights. (Round your answer to two decimal places.)

c. Find the 85th percentile for the total weight for the 100 weights. (Round your answer to two decimal places.

A simple random sample of 600 individuals provides 100 Yes responses.

a. What is the point estimate of the proportion of the population that would provide Yes responses (to 2 decimals)?

Later use p(average) rounded to 2 decimal places.

b. What is your estimate of the standard error of the proportion (to 4 decimals)?

c. Compute the 95% confidence interval for the population proportion (to 4 decimals).

The correct answers are highlighted. I would like an explanation for why some are null hypothesis and some are alternative. Thank you.

A bag of gumballs contains 3 grape balls, 4 cherry balls, and 5 lemon balls.

1. How many ways can 2 grape balls, 2 cherry balls, and 3 lemon balls be selected?

2. Determine the probability of selecting 2 grape balls, 2 cherry balls, and 3 lemon balls?

*Please show all work and clear answers, and if possible, a link to resource used to solve this problem. I've searched every resource available and I can't figure it out! Thank you in advance!*

A friend makes three pancakes for breakfast. One of the pancakes is burned on both sides, one is burned on only one side, and the other is not burned on either side. You are served one of the pancakes at random, and the side facing you is burned. What is the probability that the other side is burned? (Hint: Use conditional probability.)

Enter the exact answer.

**The answer is NOT 0.5 or 1/2, when I put either of those, it says it is incorrect.**

1. The tax officials at the Internal revenue Service (IRS) are constantly working toward improving the wording and format of the tax returns. As part of a larger effort to help taxpayers, the Internal Revenue Service plans to streamline one of the forms into a shorter and simpler form for the 2021 tax season.

2. Upon successful completion of this exercise, the new form, – about half the size of the current version – would replace the previous ones and will be shared with the tax community for the feedback. The new Form uses a “building block” approach, in which the tax return is reduced to a simple form. That form can be supplemented with additional schedules if needed. Taxpayers with straightforward tax situations would only need to file this new form with no additional schedules.

To finalize this exercise, the IRS have developed three new forms, and to determine which, if any, are superior to the current forms, 120 individuals were asked to participate in an experiment. Each of the three new forms and the currently used form were filled out by 30 different people. The amount of time (in minutes) taken by each person to complete the task was recorded. The data collected is attached in the Excel file named: Tax Forms worksheet.

You are expected to analyze the project in two phases:

Phase1:

1. a) Describe the problem background, objective of study and identify the type of scale of measurement for the data

2. b) Use appropriate descriptive statistics to explore and summarize the data for Tax form 2 & 3 and compare their results. Remember to interpret the findings accurately and present them in a clear and coherent way.

3. c) Assuming data for Form 2 is normally distributed, calculate the parentage of people who completed the form between 83.9 and 107.5 minutes (round the descriptive statistics numbers to one decimal)

4. d) If the filling time of all IRS forms is distributed normally with mean of 102 and standard deviation of 8, what is the probability that a randomly selected person could do the tax forms in less than 90 minutes?

e) Referring to problem “d” above, If a randomly selected person is in the top 5 percent of the fastest people who do the tax forms, at least how many minutes should he spent to fill out the form?

"Excel sheet numbers:

Taxpayer Form 1 ------Form 2------- Form 3 ------Form 4

1-------------- 109 ------------115 ----------126 -----------120

2-------------- 98 -------------103 ----------107 -----------108

3 -------------29 --------------27------------ 53------------- 38

4------------ 93 ---------------95 ------------103---------- 109

5------------ 62--------------- 65------------ 67------------- 64

6 -----------103------------- 107----------- 111----------- 128

7------------ 83-------------- 82------------ 101------------ 116

8------------ 122------------ 119----------- 141----------- 143

9 -------------92------------ 101----------- 105------------ 108

10------------ 107--------- 113------------- 127----------- 113

11------------ 103---------- 111------------ 111------------ 108

12------------ 54------------ 64-------------- 67-------------62

13------------ 141---------- 145----------- 142------------160

14------------ 92------------ 94------------- 95-------------102

15 ------------29 -----------32------------- 33---------------62

16----------- 83------------- 83------------ 89-------------- 86

17------------ 34 -----------36 ------------40---------------48

18 ------------83----------- 86------------- 90-------------119

19------------ 157---------- 157----------- 172-----------193

20------------- 99---------- 107------------- 111-----------100

21----------- 118----------- 123------------- 117----------130

22------------ 58----------- 65--------------- 75-------------81

23 ------------66----------- 71---------------- 79------------81

24------------ 60----------- 60--------------- 78------------ 41

25 ------------102---------- 106------------ 100---------- 142

26 -------------128---------- 134------------ 135--------- 142

27---------------87---------- 93------------- 90------------ 77

28--------------126-------- 134----------- 129----------- 154

29----------- -133---------- 130----------- 148----------- 164

30------------ 100----------- 112---------- 107----------- 120

Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α = 8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter μ = 8t. (Round your answers to three decimal places.)

(a) What is the probability that exactly 8 small aircraft arrive during a 1-hour period?

What is the probability that at least 8 small aircraft arrive during a 1-hour period?

What is the probability that at least 12 small aircraft arrive during a 1-hour period?

(b) What is the expected value and standard deviation of the number of small aircraft that arrive during a 105-min period?

(c) What is the probability that at least 23 small aircraft arrive during a 2.5-hour period?

What is the probability that at most 11 small aircraft arrive during a 2.5-hour period?