GRADED PROBLEM SET #5

Answer each of the following questions completely. There are a total of 20 points possible in the assignment.

1. (8 pts) Based on past results found in the Information Please Almanac there is a 0.1919 probability that a baseball World Series will last four games, a 0.2121 probability that it will last five games, a 0.2223 probability that it will last six games, and a 0.3737 probability that it will last seven games.
   1. Does the given information describe a probability distribution? Explain.

2. Assuming that the given information describes a probability distribution, find the mean and standard deviation for the number of games in World Series.

2. (4 pts) Men heights are normally distributed with a mean of 70 inches and a standard deviation of 3 inches. Doorframes have to be designed so that 99.9% of all men can pass under without stooping. What is the height of the doorframe?

3. (8 pts) Tree diameters in a plot of land are normally distributed with a mean of 14 inches and a standard deviation of 3.2 inches.
   1. What is the probability that an individual tree has a diameter between 13 inches and 16.3 inches?

2. What is the probability that an individual tree has a diameter less than 12 inches?

3. What is the probability that an individual tree has a diameter of at least 15 inches?

4. Find the cutoff for the 80^th percentile of tree diameters. (Provide the probability notation)

Part 1: Analyzing your College’s School Graduation Rate

You recently went through your college website and some information there got your attention. There was a claim that your college has a 77% graduation rate. You thought it would be interesting to check the validity of this statement since these days you are reading about hypothesis testing in your Statistics online course. You contacted the research department and got access to the data for the last graduation and out of 200 students 165 graduated.

To complete Case 1 please answer the following questions:

1. If you are to conduct a hypothesis test using the data above,
   1. Which test (t-test or z test) would you think be appropriate and why?

2. Is this should be a test on population proportion or mean?

3. Calculate the sample statistics (sample mean or sample proportion based on the information given) (1 mark)

2. Conduct the hypothesis test with a 0.01 level of significance. You may use ‘the percentage of graduates not equals to 0.77’ as your alternative hypothesis. Please show your work, the 5 steps as described in the textbook.

3. Construct a 95% confidence interval for the population proportion of graduates. Show your work.

The probability that a random gift box in Overwatch (PC game) has one of the character skins you want is .1. Suppose you get a gift box every game you play, and that you play until you have obtained 2 of these skins. a. What is the probability that you play until you have x boxes that do not have the desired prize? Write down the formula as well as the notation for the pdf. b. What is the probability that you play exactly 5 times? Show the R code. c. What is the probability that you play at most 5 times? Show the R code. d. How many boxes without the desired skins do you expect to get? Show the formula

Describe the sampling distribution of ModifyingAbove p with caret. Assume the size of the population is 15 comma 000. nequals200​, pequals0.4 Choose the phrase that best describes the shape of the sampling distribution of ModifyingAbove p with caret below. A. Approximately normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis less than 10. B. Not normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis less than 10. C. Approximately normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis greater than or equals 10. D. Not normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis greater than or equals 10. Determine the mean of the sampling distribution of ModifyingAbove p with caret. mu Subscript ModifyingAbove p with caret Baseline equals nothing ​(Round to one decimal place as​ needed.) Determine the standard deviation of the sampling distribution of ModifyingAbove p with caret. sigma Subscript ModifyingAbove p with caret Baseline equals nothing ​(Round to three decimal places as​ needed.)

a.) Suppose that government data show that 8% of adults are full‑time college students and that 30% of adults are age 55 or older. Complete the passage describing the relationship between the two aforementioned events. Although (0.08)⋅(0.30)=0.024, we cannot conclude that 2.4% of adults are college students 55 or older because the two events __________(are/are not) ________(independent/disjoint)

b.) In New York State's Quick Draw lottery, players choose between one and ten numbers that range from 11 to 80.80. A total of 2020winning numbers are randomly selected and displayed on a screen. If you choose a single number, your probability of selecting a winning number is 2080,2080, or 0.25.0.25. Suppose Lester plays the Quick Draw lottery 66 times. Each time, Lester only chooses a single number.

What is the probability that he loses all 66 of his lottery games? Please give your answer to three decimal places.

c.) Consider the sample space of all people living in the United States, and within that sample space, the following two events.

??=people who play tennis=people who are left‑handedA=people who play tennisB=people who are left‑handed

Suppose the following statements describe probabilities regarding these two events. Which of the statements describe conditional probabilities? Select all that apply:

-Two‑tenths of a percent of people living in the United States are left‑handed tennis players.

-Two percent of left‑handed people play tennis.

-Of people living in the United States, 3.7% play tennis.

-There is a 10.2% chance that a randomly chosen person is left‑handed.

-The probability is 5.4% that a tennis player is left‑handed.

-There is a 13.7% probability that a person is a tennis player or left‑handed.

d.)

Of all college degrees awarded in the United States, 50%50% are bachelor's degrees, 59%59% are earned by women, and 29%29% are bachelor's degrees earned by women. Let ?(?)P(B) represent the probability that a randomly selected college degree is a bachelor's degree, and let ?(?)P(W) represent the probability that a randomly selected college degree was earned by a woman.

What is the conditional probability that a degree is earned by a woman, given that the degree is a bachelor's degree? Please round your answer to the two decimal places.

The Bogard Corporation produces three types of bookcases, which it sells to large office supply companies. The production of each bookcase requires two machine operations, trimming and shaping, followed by assembly, which includes inspection and packaging. Each type requires 0.4 hours of assembly time, but the machining operations have different processing times, as shown in the table below (in hours per unit). Each machine is available for 150 hours per month, and the current size of the assembly department provides capacity of 200 hours. Each bookcase produced yields a unit profit contribution as shown below.

Standard Narrow Wide

Trimmer 0.2 0.4 0.6

Shaper 0.6 0.2 0.5

Profit $8 $6 $10

Write a linear optimization model (i.e., identify decision variables, objective function and constraints)

#37

Let x = age in years of a rural Quebec woman at the time of her first marriage. In the year 1941, the population variance of x was approximately σ² = 5.1. Suppose a recent study of age at first marriage for a random sample of 31 women in rural Quebec gave a sample variance s² = 2.3. Use a 5% level of significance to test the claim that the current variance is less than 5.1. Find a 90% confidence interval for the population variance.

(a) What is the level of significance?


State the null and alternate hypotheses.

H_o: σ² = 5.1; H₁: σ² < 5.1

H_o: σ² < 5.1; H₁: σ² = 5.1  

  H_o: σ² = 5.1; H₁: σ² ≠ 5.1

H_o: σ² = 5.1; H₁: σ² > 5.1

(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)


What are the degrees of freedom?


What assumptions are you making about the original distribution?

We assume a normal population distribution.

We assume a uniform population distribution.  

  We assume a binomial population distribution.

We assume a exponential population distribution.

(c) Find or estimate the P-value of the sample test statistic.

P-value > 0.100

0.050 < P-value < 0.100  

  0.025 < P-value < 0.050

0.010 < P-value < 0.025

0.005 < P-value < 0.010

P-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

Since the P-value > α, we fail to reject the null hypothesis.Since the P-value > α, we reject the null hypothesis.

   Since the P-value ≤ α, we reject the null hypothesis.Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 5% level of significance, there is insufficient evidence to conclude that the variance of age at first marriage is less than 5.1.

At the 5% level of significance, there is sufficient evidence to conclude that the that the variance of age at first marriage is less than 5.1.    

(f) Find the requested confidence interval for the population variance. (Round your answers to two decimal places.)

Interpret the results in the context of the application.

We are 90% confident that σ² lies below this interval.We are 90% confident that σ² lies within this interval.  

  We are 90% confident that σ² lies above this interval.We are 90% confident that σ² lies outside this interval.

As part of Office Depot’s cost reducing strategy, research is being conducted on its line of printers. Currently Office Depot offers 5 types of printers, each having different features. Office Depot detected that when it comes to buying a printer, customers look first for a particular printing speed (pages per minute, or ppm) and only then they look to the price to make their decision. A regression analysis is performed to determine whether printer speed is a driver of (explains) the price. The following table displays the printers, their speeds and associated prices.

1. Calculate coefficient B1
2. Calculate coefficient Bo
3. The regression equation for this problem is......
4. Provide an interpretation for the intercept coefficient.
5. Provide an interpretation for the slope coefficient.

Show all work. Use Minitab as much as you can.

Majority of problems from: Montgomery & Runger, 2007

Question 1:

The following data are the joint temperatures of the O-rings (degrees F) for each test firing or actual launch of the space shuttle rocket motor: 84, 49, 61, 40, 83, 67, 45, 66, 70, 69, 80, 58, 68, 60, 67, 72, 73, 70, 57, 63, 70, 78, 52, 67, 53, 67, 75, 61, 70, 81, 76, 79, 75, 76, 58, 31.

1. Compute the sample mean and sample standard deviation of the temperature data.
2. Graph a histogram of the data.
3. Graph a normal probability plot of the data.
4. Comment on the data

Question 2:

The mean water temperature downstream from a power plant cooling tower discharge pipe should be no more than 100 degrees F. Past experience has indicated that the standard deviation of temperature is 2 degrees F. The water temperature is measured on nine randomly chosen days, and the average temperature is found to be 98 degrees F.

1. Is there evidence that the water temperature is acceptable at α =0.05?
2. What is the p-value for this test?

Question 3:

Suppose that 1000 customers are surveyed and 850 are satisfied or very satisfied with a corporation’s products and services.

1. Test the hypothesis H₀: p = 0.9 against H₁: p ≠ 0.9 at α = 0.05. Find the p-value.
2. Explain how the question in part (a) could be answered by constructing a 95% two-sided confidence interval for p.

Consider the following sample data for the relationship between advertising budget and sales for Product A: Observation 1 2 3 4 5 6 7 8 9 10 Advertising ($) 40,000 50,000 50,000 60,000 70,000 70,000 80,000 80,000 90,000 100,000 Sales ($) 240,000 308,000 315,000 358,000 425,000 440,000 499,000 494,000 536,000 604,000 What is the slope of the "least-squares" best-fit regression line? Please round your answer to the nearest hundredth. Note that the correct answer will be evaluated based on the full-precision result you would obtain using Excel.

. The state's education secretary claims that the average cost of one year's tuition for all private high schools in the state is $2350.00.A sample of 35 private high schools is selected, and the average tuition is $2315.00. The population standard deviation is $38.00. At a significance level of 0.05, is there enough evidence to reject the claim that the average cost of tuition is equal to $2350.00?

Steel rods are manufactured with a mean length of 24 centimeter​ (cm). Because of variability in the manufacturing​ process, the lengths of the rods are approximately normally distributed with a standard deviation of 0.07 cm. Complete parts ​(a) to ​(d). ​(a) What proportion of rods has a length less than 23.9 ​cm? 0.0764 ​(Round to four decimal places as​ needed.) ​(b) Any rods that are shorter than 23.84 cm or longer than 24.16 cm are discarded. What proportion of rods will be​ discarded? nothing ​(Round to four decimal places as​ needed.)

Abstract

This study investigated associations between working memory (measured by complex memory tasks) and both reading and mathematics abilities, as well as the possible mediating factors of fluid intelligence, verbal abilities, short-term memory (STM), and phonological awareness, in a sample of 6- to 11-year-olds with reading disabilities. As a whole, the sample was characterized by deficits in complex memory and visuospatial STM and by low IQ scores; language, phonological STM, and phonological awareness abilities fell in the low average range. Severity of reading difficulties within the sample was significantly associated with complex memory, language, and phonological awareness abilities, whereas poor mathematics abilities were linked with complex memory, phonological STM, and phonological awareness scores. These findings suggest that working memory skills indexed by complex memory tasks represent an important constraint on the acquisition of skill and knowledge in reading and mathematics. Possible mechanisms for the contribution of working memory to learning, and the implications for educational practice, are considered.

Citation:Gathercole, S. E., Alloway, T. P., Willis, C., & Adams, A. M. (2006). Working memory in children with reading disabilities. Journal of Experimental Child Psychology, 93(3), 265-281.

Dataset:

-  Dependent variable (Y): Reading - reading skills of the 6 to 11 year olds
-   Independent variables (X):
    - Verbal - a measure of verbal ability (spelling, phonetics, etc.)
    - Math - a measure of math ability
    - Work_mem - working memory score

Data screening:

Accuracy

Assume the data is accurate with no missing values. You will want to screen the dataset using all the predictor variables to predict the outcome in a simultaneous multiple regression (all the variables at once). This analysis will let you screen for outliers and assumptions across all subsequent analyses/steps.

Outliers

a. Leverage:
    i.  What is your leverage cut off score?
    ii. How many leverage outliers did you have?



b.  Cook's:
    i.  What is your Cook's cut off score?
    ii. How many Cook's outliers did you have?
    


c.  Mahalanobis:
    i.  What is your Mahalanobis df?
    ii. What is your Mahalanobis cut off score?
    iii.    How many outliers did you have for Mahalanobis?
    


d.  Overall:
    i.  How many total outliers did you have across all variables?
    ii. Delete them!

Hierarchical Regression:

a. In step 1, control for verbal ability of the participant predicting reading scores.
b.  In step 2, test if working memory is related to reading scores.
c.  In step 3, test if math score is related to reading scores.
d.  Include the summaries of each step, along with the ANOVA of the change between each step.

Moderation:

a. Examine the interaction between verbal and math scores predicting reading scores.
b. Include the simple slopes for low, average, and high math levels (split on math) for verbal predicting reading.
c. Include a graph of the interaction.

A transect is an archaeological study area that is 1/5 mile wide and 1 mile long. A site in a transect is the location of a significant archaeological find. Let x represent the number of sites per transect. In a section of Chaco Canyon, a large number of transects showed that x has a population variance σ² = 42.3. In a different section of Chaco Canyon, a random sample of 18 transects gave a sample variance s² = 49.7 for the number of sites per transect. Use a 5% level of significance to test the claim that the variance in the new section is greater than 42.3. Find a 95% confidence interval for the population variance.

(a) What is the level of significance?


State the null and alternate hypotheses.

H_o: σ² = 42.3; H₁: σ² ≠ 42.3

H_o: σ² > 42.3; H₁: σ² = 42.3  

  H_o: σ² = 42.3; H₁: σ² > 42.3

H_o: σ² = 42.3; H₁: σ² < 42.3

(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)


What are the degrees of freedom?


What assumptions are you making about the original distribution?

We assume a uniform population distribution.We assume a binomial population distribution.    We assume a normal population distribution.We assume a exponential population distribution.

(c) Find or estimate the P-value of the sample test statistic.

P-value > 0.100

0.050 < P-value < 0.100

   0.025 < P-value < 0.050

0.010 < P-value < 0.025

0.005 < P-value < 0.010

P-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

Since the P-value > α, we fail to reject the null hypothesis.

Since the P-value > α, we reject the null hypothesis.

   Since the P-value ≤ α, we reject the null hypothesis

.Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 5% level of significance, there is insufficient evidence to conclude conclude that the variance is greater in the new section.

At the 5% level of significance, there is sufficient evidence to conclude conclude that the variance is greater in the new section.    

(f) Find the requested confidence interval for the population variance. (Round your answers to two decimal places.)

Interpret the results in the context of the application.

We are 95% confident that σ² lies within this interval.

We are 95% confident that σ² lies below this interval.  

  We are 95% confident that σ² lies above this interval

.We are 95% confident that σ² lies outside this interval.

According to a survey, 21% of the car owners said that they get the maintenance service done on their cars according to the schedule recommended by the auto company. Suppose that this result is true for the current population of car owners.

Find to 3 decimal places the probability that exactly 4 car owners in a random sample of 13 get the maintenance service done on their cars according to the schedule recommended by the auto company. Use the binomial probability distribution formula.