The mean SAT score in mathematics,

μ

, is

559

. The standard deviation of these scores is

39

. A special preparation course claims that its graduates will score higher, on average, than the mean score

559

. A random sample of

50

students completed the course, and their mean SAT score in mathematics was

561

. At the

0.05

level of significance, can we conclude that the preparation course does what it claims? Assume that the standard deviation of the scores of course graduates is also

39

.

Perform a one-tailed test. Then fill in the table below.

Carry your intermediate computations to at least three decimal places, and round your responses as specified in the table. (If necessary, consult a list of formulas.)

23.4Matching Dogs and Owners. Researchers constructed two test sheets, each sheet including 20 photos of the faces of dog-owner pairs taken at a dog-lovers field festival. The 20 sets of dog-owner pairs on the two sheets were equivalent with respect to breed, diversity of appearance, and gender of owners. On the first sheet, the dogs were matched with their owners, while on the second sheet, the dogs and owners were deliberately mismatched. Three experiments were conducted, and in all experiments, subjects were asked to “choose the set of dog-owner pairs that resemble each other, Sheet 1 or Sheet 2,” and were simply told the aim of the research was a “survey on dog-owner relationships.” In the first experiment, the original sheets were shown to subjects; in the second experiment, just the “mouth region” of the owners was blacked out in all the pictures on both sheets; while in the third experiment, just the “eye region” of the owners was blacked out. Subjects were assigned at random to the three experimental groups, and in each experiment, the number of subjects who selected the sheet with the dogs and their owners correctly matched was recorded. Experimenters were interested in whether blacking out portions of the face reduced the ability of subjects to correctly match dogs and owners.⁸ Here are the results:

1. Is there evidence that blacking out the mouth reduces a subject’s ability to choose the sheet which correctly matches the dogs and their owners? Follow the four-step process as exhibited in Examples 23.4 and 23.5.
2. Is there evidence that blacking out the eyes reduces a subject’s ability to choose the sheet which correctly matches the dogs and their owners? Follow the four-step process as exhibited in Examples 23.4 and 23.5.
3. Contrast your conclusions in parts (a) and (b) in the context of the problem using non-technical language.

Students in a business statistics course performed a completely randomized design to test the strength of four brands of trash bags. One-pound weights were placed into a bag, one at a time, until the bag broke. A total of 40 bags, 10 for each brand, were used. The data in give the weight (in pounds) required to break the trash bags. Dataset: Trashbags

1. At the 0.05 level of significance, is there evidence of a difference in the mean strength of the four brands of trash bags?
2. If appropriate, determine which brands differ in mean strength.
3. At the 0.05 level of significance, is there evidence of a difference in the variation in strength among the four brands of trash bags?

   Kroger	Glad	Hefty	TuffStuff
   34	32	33	26
   30	42	34	18
   40	34	32	20
   38	36	40	15
   36	32	40	20
   30	40	34	20
   30	36	36	17
   42	43	34	18
   36	30	32	19
   38	38	34	20

Diana's kids, Naomi and Isaac, play a lot of video games together. On a particular level, the number of points Naomi scores has a Discrete Uniform distribution on [5,11] and the number of points Isaac scores has a Discrete Uniform distribution on [6, 9]. Each time they play, their scores are independent.

(a) (2) Write down an expression for the probability that in 100 games, the total number of points scored by Naomi and Isaac is at least 1600. Do not evaluate the expression.

(b) (2) Find the mean and variance of the total number of points scored by Naomi and Isaac in one game.

(c) (6) Using the Central Limit Theorem, find the approximate probability in (a). You should use a continuity correction. Justify why the approximation is appropriate in this situation.

(d) (5) Suppose Naomi and Isaac want to play until they have a 96% probability of being above a total score of 1600. What is the smallest number of games they must play to achieve this?

1) One of the biggest flaws in any research is the logic of statistical correlation implies causation. What is the fundamental difficulty with this assumption and how should correlations be treated?

2) Explain the difference between discrete and continuous data. Provide an example of each.

3) What is the difference between an observational study and an experiment?

4) What is a histogram and what is its value to statistical presentations?

5) What is a frequency distribution and why is it useful?

6) Describe at least 3 sampling methods discussed in the chapter. What are their strengths and weaknesses?

Directions: your answers should be typed to 200-300 words

A research facility is evaluating five different processes of extracting starch from corn flour.

The percent yield of starch is the response of interest. The data is shown below:

Process   Starch_Yield
1   75.9
1   72.3
1   64.7
1   68.7
1   73.4
1   77.7
2   77.7
2   74.9
2   78.3
2   77.3
3   63.5
3   60.4
3   65.3
3   62.6
3   59.3
4   61
4   64
4   55.9
5   76.3
5   75.3
5   71.5
5   81

a) Perform an ANOVA test and report the F value and use 2 decimal places

b) Are the processes significantly different?

c) Which process(es) would you pick?

Solve using R studio and post code

As a student you have probably noticed a curious phenomenon. In every class, there are some students who zip through exams and turn their papers in while everyone else is still working. Other students continue working until the very last minute. Have you ever wondered what grades these students get? Are the students who finish first the best in the class or are they simply conceding failure? To answer this question, we carefully observed a recent exam and recorded the amount of time each student spent working (X) and the grade they received (Y). The data from the sample of n = 10 students is below.

a) compute the Pearson correlation to measure the degree of relationship between the time spent writing the exam and the grade. Is the correlation statistically significant? State the null hypothesis, use α = .05 two-tailed and include a summary statement.

b) What percentage of variance in grades is predicted from time spent writing the exam?

The life span of a calculator manufactured by Texas Instruments has a normal distribution with a mean of 65 months and a standard deviation of 1 year. The company guarantees that any calculator that starts malfunctioning within 38 months of the purchase will be replaced by a new one. About what percentage of calculators made by this company are expected to be replaced? (Hint: Draw a picture, shade the region, label the axes.) If the company sells 5000 calculators, how many are expected to be replaced?

The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures the motivation, attitude toward school, and study habits of students. Scores range from 0 to 200. A selective private college gives the SSHA to an SRS of both male and female first-year students. The data for the women are as follows: 154 109 137 115 152 140 154 178 101 103 126 126 137 165 165 129 200 148 Here are the scores of the men: 108 140 114 91 180 115 126 92 169 146 109 132 75 88 113 151 70 115 187 104 Most studies have found that the mean SSHA score for men is lower than the mean score in a comparable group of women. Is this true for first-year students at this college? Use a 1% significance level

(a) Hypotheses and results:

(b) Draw a picture and label p-value and horizontal axis:

(c) Draw a conclusion. Don’t just accept or reject. Say what it means in terms of this problem.

4. Core i5 and i7 are two different types of CPU manufactured by Intel. As you may know as a matter of fact, Intel does not produce two types of CPUs. Instead, they just produce Core i7 chips. However, since a chip contains many millions of transistors, some of the transistors may not work properly, while other regions from the chip are working perfectly. In this case, Intel does not scrap the chip to trash, as it will be a waste, but deactivates the malfunctioning region and sell it as a cheaper chip, named Core i5, with less but perfectly functional features. Suppose an assembly line is able to manufacture 3000 chips per day. The probability that a chip meets the Core i7 standard is independently 1/1000. (While the background of this question is real, the probability here is much lower than the true probability.) An assembly line is qualified if it is able to produce at least 3 Core i7 per day. a) Write down the exact expression of the probability that this assembly line is qualified if it operates only one day. (b) Write down a relevant approximate expression for the probability from (a).

5. Consider the assembly line described in the above question. Since Core i7 makes much more profit than Core i5, Intel decides to manufacture more chips in a day in order to produce sufficient Core i7 chips per day with high probability. Specifically, the CEO wants to get at least 3 Core i7 chips per day with probability at least 0.999. How many assembly lines should Intel purchase in total? (You might need wolframalpha.com or other devices to solve an equation.)

A small business ships specialty homemade candies to anywhere in the world. Past records indicate that the weight of orders is normally distributed. Suppose a random sample of 16 orders is selected and each is weighed. The sample mean was found to be 110 grams with a standard deviation of 14 grams. Show your work

A. Describe the sampling distribution for the sample mean.

B. What is the standard error?

C.. For 90% confidence, what is the margin of error?

D. Based on the sample results, create the 90% confidence interval and interpret.

Data:

sample   volume
1 15.5
2 16.6
3 15.5
4 16.1
5 14.4

A bottling company claims that the mean amount of fluid in its beverage bottles is 16 oz. It's known that the amount of fluid is normally distributed. The amount of fluid of a sample of 9 nine bottles is shown above:

a) Test the company's claim at a .05 level of significance What is the value of the statistic?

b) Calculate a 94% Lower Confidence bound (1-sided)  for the mean amount of fluid.

c)  What minimum sample size is needed to detect a shift equal to 1 standard deviation, if the required power is .99 and the level of significance is .01

Assume we want a two-sided test.

Solve using R studio, must include code

6   15.7
7   15.4
8   14.5
9   15.5

A random sample of 100 households and collect data on each household’s income. You find a sample income standard deviation of $160,000. You want to test the null hypothesis that the population mean income is $100,000, using a p-value of 0.05, against the alternative that it does not equal $100,000.

You will:

A. Reject the null hypothesis if the sample mean is $130,000

B. Fail to reject the null hypothesis if the sample mean is between $80,000 and $120,000

C. Fail to reject the null hypothesis if the sample mean is less than $120,000

D. Accept the null hypothesis

Let X₁ and X₂ be two observations on an iid random variable drawn from a population with mean μ and variance σ² (for example, the random variable could be income; X₁ could be person 1’s income and X₂, person 2’s income). Which of the following is an unbiased estimator of mean income?

All of these are unbiased estimators of mean income.

23X1+13X2

15X1+45X2

Which of the following is the “best” (that is, least-variance) estimator of mean income?

15X1+45X2

23X1+13X2

All of these are "best" estimators of mean income.