You may need to use the appropriate technology to answer this question.

Health insurance benefits vary by the size of the company. The sample data below show the number of companies providing health insurance for small, medium, and large companies. For purposes of this study, small companies are companies that have fewer than 100 employees. Medium-sized companies have 100 to 999 employees, and large companies have 1,000 or more employees.

The questionnaire sent to 225 employees asked whether or not the employee had health insurance and then asked the employee to indicate the size of the company.

(a)

Conduct a test of independence to determine whether health insurance coverage is independent of the size of the company.

State the null and alternative hypotheses.

H₀: Having health insurance is not independent of the size of the company.
H_a: Having health insurance is independent of the size of the company. H₀: Having health insurance is not independent of the size of the company.
H_a: The proportion of companies with health insurance benefits is not equal for small, medium and large companies.     H₀: Having health insurance is independent of the size of the company.
H_a: The proportion of companies with health insurance benefits is equal for small, medium and large companies.H₀: Having health insurance is independent of the size of the company.
H_a: Having health insurance is not independent of the size of the company.

Find the value of the test statistic. (Round your answer to three decimal places.)

What is the p-value? (Round your answer to four decimal places.)

p-value =

Using a 0.05 level of significance, what is your conclusion?

Do not reject H₀. We cannot conclude health insurance coverage and the size of the company are not independent.Do not reject H₀. We cannot conclude health insurance coverage is independent of the size of the company.     Reject H₀. We conclude health insurance coverage is independent of the size of the company. Reject H₀. We conclude health insurance coverage is not independent of the size of the company.

(b)

A newspaper article indicated employees of small companies are more likely to lack health insurance coverage. Use percentages based on the above data to support this conclusion. (Round your answers to the nearest integer.)

For small companies,  % do not provide health insurance. For medium companies,  % do not provide health insurance. For large companies,  % do not provide health insurance. These percentages support the conclusion that small companies are  ---Select--- less more equally likely to provide health insurance coverage when compared to medium and large companies.

A population consists of the following N=6 scores: 2,4,1,2,7,2 a) compute mean and standard deviation for the population b)Find z-score for each score in the population c) Transform the original population into a new population of N=5 scores with a mean of M(mu)=50 and a standard deviation of sigma=10?

You may need to use the appropriate technology to answer this question.

A standardized exam consists of three parts: math, writing, and critical reading. Sample data showing the math and writing scores for a sample of 12 students who took the exam follow.

(a)

Use a 0.05 level of significance and test for a difference between the population mean for the math scores and the population mean for the writing scores. (Use math score − writing score.)

Formulate the hypotheses.

H₀: μ_d > 0

H_a: μ_d ≤ 0

H₀: μ_d ≤ 0

H_a: μ_d > 0

H₀: μ_d ≠ 0

H_a: μ_d = 0

H₀: μ_d = 0

H_a: μ_d ≠ 0

H₀: μ_d ≤ 0

H_a: μ_d = 0

Calculate the test statistic. (Round your answer to three decimal places.)

Calculate the p-value. (Round your answer to four decimal places.)

p-value =

What is your conclusion?

Do not reject H₀. We cannot conclude that there is a significant difference between the population mean scores for the math test and the writing test. Reject H₀. We cannot conclude that there is a significant difference between the population mean scores for the math test and the writing test.      Reject H₀. We can conclude that there is a significant difference between the population mean scores for the math test and the writing test. Do not reject H₀. We can conclude that there is a significant difference between the population mean scores for the math test and the writing test.

(b)

What is the point estimate of the difference between the mean scores for the two tests? (Use math score − writing score.)

What are the estimates of the population mean scores for the two tests?

Math Writing

Which test reports the higher mean score?

The math test reports a  ---Select--- higher lower mean score than the writing test.

Filling boxes with consistent amounts of its cereals is critical to General Mills's success. The filling variance for boxes of Count Chocula cereal is designed to be 0.02 ounces² or less. A sample of 31 boxes of Count Chocula shows a sample standard deviation of 0.16 ounces. Use α = 0.05 to determine whether the variance in the cereal box fillings is exceeding the design specification.

A) State the null and alternative hypotheses.

a) H₀: σ² = 0.02

H_a: σ² ≠ 0.02

b) H₀: σ² < 0.02

H_a: σ² ≥ 0.02

c) H₀: σ² ≥ 0.02

H_a: σ² < 0.02

d) H₀: σ² ≤ 0.02

H_a: σ² > 0.02

e) H₀: σ² > 0.02

H_a: σ² ≤ 0.02

B) Find the value of the test statistic.

test statistic=

Find the p-value. (Round your answer to four decimal places.)

p-value =

State your conclusion.

Reject H₀. The population variance does appear to be exceeding the standard.

Reject H₀. The population variance does not appear to be exceeding the standard.    

Do not reject H₀. The population variance does appear to be exceeding the standard.

Do not reject H₀. The population variance does not appear to be exceeding the standard

You may need to use the appropriate technology to answer this question.

Based on sales over a six-month period, the five top-selling compact cars are Chevy Cruze, Ford Focus, Hyundai Elantra, Honda Civic, and Toyota Corolla.† Based on total sales, the market shares for these five compact cars were Chevy Cruze 24%, Ford Focus 21%, Hyundai Elantra 20%, Honda Civic 18%, and Toyota Corolla 17%. Suppose a sample of 400 compact car sales in one city showed the following number of vehicles sold.

Use a goodness of fit test to determine if the sample data indicate that the market shares for the five compact cars in this city are different than the market shares reported by Motor Trend. Use a 0.05 level of significance.

State the null and alternative hypotheses.

H₀: The majority of the market shares for the five automobiles in this city differ from the ones reported by Motor Trend.
H_a: The majority of the market shares for the five automobiles in this city are the same as the ones reported by Motor Trend.H₀: The market shares for the five automobiles in this city differ from 0.24, 0.21, 0.20, 0.18, 0.17.
H_a: The market shares for the five automobiles in this city are the same as the above shares.     H₀: The majority of the market shares for the five automobiles in this city are the same as the ones reported by Motor Trend.
H_a: The majority of the market shares for the five automobiles in this city differ from the ones reported by Motor Trend.H₀: The market shares for the five automobiles in this city are 0.24, 0.21, 0.20, 0.18, 0.17.
H_a: The market shares for the five automobiles in this city differ from the above shares.

Find the value of the test statistic. (Round your answer to three decimal places.)

Find the critical value for the test. (Round your answer to three decimal places.)

critical value =

State your conclusion.

Reject H₀. We cannot conclude that the market shares for the five compact cars in this city differ from the market shares reported. Do not reject H₀. We conclude that the market shares for the five compact cars in this city differ from the market shares reported.      Do not reject H₀. We cannot conclude that the market shares for the five compact cars in this city differ from the market shares reported. Reject H₀. We conclude that the market shares for the five compact cars in this city differ from the market shares reported.

What market share differences, if any, exist in this city?

Chevy Cruze shows  ---Select--- a higher a lower the same market share in this city. Ford Focus shows  ---Select--- a higher a lower the same market share in this city. Hyundai Elantra shows  ---Select--- a higher a lower the same market share in this city. Honda Civic shows  ---Select--- a higher a lower the same market share in this city. Toyota Corolla shows  ---Select--- a higher a lower the same market share in this city.

You may need to use the appropriate technology to answer this question.

The College Board provided comparisons of Scholastic Aptitude Test (SAT) scores based on the highest level of education attained by the test taker's parents. A research hypothesis was that students whose parents had attained a higher level of education would on average score higher on the SAT. The overall mean SAT math score was 514.† SAT math scores for independent samples of students follow. The first sample shows the SAT math test scores for students whose parents are college graduates with a bachelor's degree. The second sample shows the SAT math test scores for students whose parents are high school graduates but do not have a college degree.

(a)

Formulate the hypotheses that can be used to determine whether the sample data support the hypothesis that students show a higher population mean math score on the SAT if their parents attained a higher level of education. (Let μ₁ = population mean verbal score of students whose parents are college graduates with a bachelor's degree and μ₂ = population mean verbal score of students whose parents are high school graduates but do not have a college degree.) For purposes of this study, assume the population variances are unequal when conducting the t-test.

H₀: μ₁ − μ₂ < 0

H_a: μ₁ − μ₂ = 0

H₀: μ₁ − μ₂ = 0

H_a: μ₁ − μ₂ ≠ 0

H₀: μ₁ − μ₂ ≠ 0

H_a: μ₁ − μ₂ = 0

H₀: μ₁ − μ₂ ≥ 0

H_a: μ₁ − μ₂ < 0

H₀: μ₁ − μ₂ = 0

H_a: μ₁ − μ₂ > 0

(b)

What is the point estimate of the difference between the means for the two populations?

(c)

Find the value of the test statistic. (Round your answer to three decimal places.)

Compute the p-value for the hypothesis test. (Round your answer to four decimal places.)

p-value =

(d)

At

α = 0.05,

what is your conclusion?

Do not reject H₀. There is insufficient evidence to conclude that higher population mean verbal scores are associated with students whose parents are college graduates.Do not Reject H₀. There is sufficient evidence to conclude that higher population mean verbal scores are associated with students whose parents are college graduates.      Reject H₀. There is insufficient evidence to conclude that higher population mean verbal scores are associated with students whose parents are college graduates.Reject H₀. There is sufficient evidence to conclude that higher population mean verbal scores are associated with students whose parents are college graduates.

5. Arsalaan A., a well-known financial analyst, selected 50 consecutive years of U.S. financial markets data at random. For 11 of the years, the rate of return for the Dow Jones Industrial Average [DJIA] exceeded the rates of return for both the S&P 500 Index and the NASDAQ Composite Index. For 8 of the years, the rate of return for the DJIA trailed the rates of return for both the S&P 500 and the NASDAQ. For 21 of the years, the rate of return for the DJIA trailed the rate of return for the S&P 500. Over the 50 years,

a. determine the probability the rate of return for the DJIA trailed the rate of return for the NASDAQ.

b. determine the probability the rate of return for the DJIA trailed the rate of return for at least one of the other two Indexes.

c. determine the probability the rate of return for the DJIA trailed the rate of return for the S&P 500 given it trailed the rate of return for the NASDAQ.

d. determine the probability the rate of return for the DJIA exceeded the rate of return for the S&P 500 given it exceeded the rate of return for the NASDAQ

1. A textbook has 500 pages on which typographical errors could occur. Suppose that there are exactly 10 such errors randomly located on those pages. Find the probability that a random selection of 50 pages will contain no errors. Find the probability that 50 randomly selected pages will contain at least two errors.

A report states that adults 18- to 24- years-old send and receive 128 texts every day. Suppose we take a sample of 25- to 34- year-olds to see if their mean number of daily texts differs from the mean for 18- to 24- year-olds.

(a) State the null and alternative hypotheses we should use to test whether the population mean daily number of texts for 25- to 34-year-olds differs from the population daily mean number of texts for 18- to 24-year-olds. (Enter != for ≠ as needed.)

H_0:

H_a:

(b) Suppose a sample of thirty 25- to 34-year-olds showed a sample mean of 118.9 texts per day. Assume a population standard deviation of 33.17 texts per day.

Compute the p-value. (Round your answer to four decimal places.)

p-value =

(c) With α = 0.05 as the level of significance, what is your conclusion?

(d) Repeat the preceding hypothesis test using the critical value approach.

State the null and alternative hypotheses. (Enter != for ≠ as needed.)

H₀:

H_a:

(e) Find the value of the test statistic. (Round your answer to two decimal places.)

State the critical values for the rejection rule. (Use α = 0.05. Round your answer to two decimal places. If the test is one-tailed, enter NONE for the unused tail.)

test statistic≤

test statistic≥

(f)

State your conclusion.

Do not reject H₀. We cannot conclude that the population mean daily texts for 25- to 34-year-olds differs significantly from the population mean of 128 daily texts for 18- 24-year-olds.

Reject H₀. We cannot conclude that the population mean daily texts for 25- to 34-year-olds differs significantly from the population mean of 128 daily texts for 18- 24-year-olds.   

Do not reject H₀. We can conclude that the population mean daily texts for 25- to 34-year-olds differs significantly from the population mean of 128 daily texts for 18- 24-year-olds.

Reject H₀. We can conclude that the population mean daily texts for 25- to 34-year-olds differs significantly from the population mean of 128 daily texts for 18- 24-year-olds.

Problem 15. Give an example of a two mutually exclusive events.

Problem 16. Give an example of three events E, F, and G so that each pair of events is mutually exclusive

Problem 17. Consider a situation where #(all) = 100, #(E) = 32, #(F) = 52, and #(E ∩ F) = 13. 1. Find P(E | F). 2. Calculate #(E ∩ F) #(F) and explain why this matches the value in part 1. Problem 18. Suppose we have 30 shuffled cards numbered 1-30. What is the probability of drawing an even value given that the value is greater than 9?

Problem 19. Suppose we roll a 6-sided die two times. What is the probability of the sum of the values being greater than 7 given that the first roll was a 5?

Discussion: Central Tendency and Variability Understanding descriptive statistics and their variability is a fundamental aspect of statistical analysis. On their own, descriptive statistics tell us how frequently an observation occurs, what is considered “average”, and how far data in our sample deviate from being “average.” With descriptive statistics, we are able to provide a summary of characteristics from both large and small datasets. In addition to the valuable information they provide on their own, measures of central tendency and variability become important components in many of the statistical tests that we will cover. Therefore, we can think about central tendency and variability as the cornerstone to the quantitative structure we are building. For this Discussion, you will examine central tendency and variability based on two separate variables. You will also explore the implications for positive social change based on the results of the data. To prepare for this Discussion: Review this week’s Learning Resources and the Descriptive Statistics media program. For additional support, review the Skill Builder: Visual Displays for Categorical Variables and the Skill Builder: Visual Displays for Continuous Variables, which you can find by navigating back to your Blackboard Course Home Page. From there, locate the Skill Builder link in the left navigation pane. Review the Chapter 4 of the Wagner text and the examples in the SPSS software related to central tendency and variability. From the General Social Survey dataset found in this week’s Learning Resources, use the SPSS software and choose one continuous and one categorical variable Note: this dataset will be different from your Assignment dataset). As you review, consider the implications for positive social change based on the results of your data.

For your continuous variable: 1. Report the mean, median, and mode. 2. What might be the better measure for central tendency? (i.e., mean, median, or mode) and why? 3. Report the standard deviation. 4. How variable are the data? 5. How would you describe this data? 6. What sort of research question would this variable help answer that might inform social change? Post the following information for your categorical variable: 1. A frequency distribution. 2. An appropriate measure of variation. 3. How variable are the data? 4. How would you describe this data? 5. What sort of research question would this variable help answer that might inform social change?

Prove that if two of the opposite sides of a quadrilateral are respectively the greatest and the least sides of the quadrilateral, then the angles adjacent to the least are greater than their opposite angles.

Color blindness is the decreased ability to see colors and clearly distinguish different colors of the visible spectrum. It is known that color blindness affects 11% of the European population. A sample of 12 subjects is taken for research purposes by a team of optometrists.

What is the probability that one subject is color blind?

What is the probability that more than one subject is color blind?

What is the probability that at most 3 subjects are color blind?

What is the probability that the number of color blind subjects is between 2 and 4 (both included)?

What is the probability that at least 10 subjects are not color blind?

Let A and B two events defined on a sample space . Show that A and B

are independent if and only if their characteristic functions 1A and 1B are independent random variables.

b. An investor wants to create a portfolio of three stocks form a set that consists of 5 defense companies, 4 transportation companies and 6 healthcare companies. If the investor picks the three stocks at random, what is the probability that all three stocks will be from different industries.

c. In part (b), what is the probability that at least two out of the three stocks picked will be from different industries?