2. Suppose we have the hypothesis test

H0 : µ = 200

Ha : µ > 200

in which the random variable X is N(µ, 10000). Let the critical region C = {x : x ≥ c}.

Find the values of n and c so that the significance level of this test is α = 0.03 and the power of µ = 220 is 0.96.

Suppose U.S. consumers 21 years and older consumed 26.2 gallons of beer and cider per person during 2017. A distributor in Milwaukee believes that beer and cider consumption are higher in that city. A sample of consumers 21 years and older in Milwaukee will be taken, and the sample mean 2017 beer and cider consumption will be used to test the following null and alternative hypotheses:

H₀: μ ≤ 26.2

H_a: μ > 26.2

(a)

Assume the sample data led to rejection of the null hypothesis. What would be your conclusion about consumption of beer and cider in Milwaukee?

Conclude that the population mean annual consumption of beer and cider in Milwaukee is greater than 26.2 gallons and hence lower than throughout the United States.

Conclude that the population mean annual consumption of beer and cider in Milwaukee is greater than 26.2 gallons and hence higher than throughout the United States.    

Conclude that the population mean annual consumption of beer and cider in Milwaukee is less than or equal to 26.2 gallons and hence lower than throughout the United States.

Conclude that the population mean annual consumption of beer and cider in Milwaukee is less than or equal to 26.2 gallons and hence higher than throughout the United States.

(b)

What is the Type I error in this situation? What are the consequences of making this error?

The type I error is rejecting H₀ when it is true. This error would claim the consumption in Milwaukee is greater than 26.2 when it is actually less than or equal to 26.2.

The type I error is not rejecting H₀ when it is true. This error would claim the consumption in Milwaukee is less than or equal to 26.2 when it is actually less than or equal to 26.2.    

The type I error is rejecting  H₀  when it is false. This error would claim the consumption in Milwaukee is greater than 26.2 when it is actually greater than 26.2.

The type I error is not rejecting H₀ when it is false. This error would claim the consumption in Milwaukee is less than or equal to 26.2 when it is actually greater than 26.2.

(c)

What is the Type II error in this situation? What are the consequences of making this error?

The type II error is accepting H₀ when it is false. This error would claim that the population mean annual consumption of beer and cider in Milwaukee is less than or equal to 26.2 gallons when it is not.

The type II error is not accepting H₀ when it is true. This error would claim that the population mean annual consumption of beer and cider in Milwaukee is greater than 26.2 gallons when it is not.    

The type II error is not accepting H₀ when it is false. This error would claim that the population mean annual consumption of beer and cider in Milwaukee is greater than 26.2 gallons when it is greater than 26.2.

The type II error is accepting H₀ when it is true. This error would claim that the population mean annual consumption of beer and cider in Milwaukee is less than or equal to 26.2 gallons when it is less than or equal to 26.2.

a) Explain intuitively why denseness is a desirable property of a sample set in sampling-based planning.

b) Explain intuitively why we want a low dispersion sample set in sampling-based planning.

Two cards are drawn at random from a standard deck of 52 cards. The number of aces drawn is counted. Prepare a probability distribution for this random experiment. Hint: Find the probability that no aces are drawn, exactly one ace is drawn, etc.

Q1.

1. Explain what happens to the 95 % confidence interval as the sample size increases.
2. Explain what happens to the width of the confidence interval for a 99% interval versus a 95%.

Q2. Consider the population of adult females resident in Melbourne. Our focus is on the population mean height. Assume we do not know ? (population standard deviation) or the population mean, µ. We take a sample of adult females resident in Melbourne (n=100) and calculate the sample mean height as 70 cm and the sample standard deviation as 25cm.

1. Derive the 95% confidence interval for the population mean
2. Compare your interval here and last week – which is larger and why?

dentify the appropriate statistical test for each research description:

1. Single-sample z-test

2. Single-samplet-test

3. Independent samples t-test

4. Related samples t-test (repeated measures or matched samples)

5. Correlation

An office manager wants to know if there’s a gender difference in punctuality. Over the period of one month, he records the number of days that each employee is late, to see if there’s a difference between male and female employees.

QUESTION 1

1. Solve the problem.
   
   A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If 40 different applicants are randomly selected, find the probability that their mean is above 215.

   		0.4713
   		0.3821
   		0.0287
   		0.1179

7.14286 points   

QUESTION 2

1. Solve the problem. Round the point estimate to the nearest thousandth.
   
   50 people are selected randomly from a certain population and it is found that 13 people in the sample are over 6 feet tall. What is the point estimate of the proportion of people in the population who are over 6 feet tall?

   		0.26
   		0.50
   		0.74
   		0.19

7.14286 points   

QUESTION 3

1. Use the confidence level and sample data to find a confidence interval for estimating the population μ. Round your answer to the same number of decimal places as the sample mean.
   
   A laboratory tested 83 chicken eggs and found that the mean amount of cholesterol was 233 milligrams with   milligrams. Construct a 95% confidence interval for the true mean cholesterol content, μ, of all such eggs.

   		229 mg < μ < 236 mg
   		229 mg < μ < 235 mg
   		231 mg < μ < 237 mg
   		230 mg < μ < 236 mg

7.14286 points   

QUESTION 4

1. Use the given information to find the minimum sample size required to estimate an unknown population mean μ.
   
   How many women must be randomly selected to estimate the mean weight of women in one age group. We want 90% confidence that the sample mean is within 3.7 lb of the population mean, and the population standard deviation is known to be 28 lb.

7.14286 points   

QUESTION 5

1. Solve the problem.
   
   Assume that women's heights are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. If 90 women are randomly selected, find the probability that they have a mean height between 62.9 inches and 64.0 inches.  ROUND TO 4 DECIMAL POSITIONS

7.14286 points   

QUESTION 6

1. Do one of the following, as appropriate: (a) Find the critical value z_α/2, (b) find the critical value t_α/2, (c) state that neither the normal nor the t distribution applies.
   
   90%; n =9; σ = 4.2; population appears to be very skewed.

   		Neither the normal nor the t distribution applies.
   		zα/2 = 1.645
   		zα/2 = 2.306
   		tα/2 = 1.860

7.14286 points   

QUESTION 7

1. Use the given degree of confidence and sample data to construct a confidence interval for the population mean μ. Assume that the population has a normal distribution.
   
   n = 30,   = 83.1, s = 6.4, 90% confidence

   		81.13 < μ < 85.07
   		81.11 < μ < 85.09
   		80.71 < μ < 85.49
   		79.88 < μ < 86.32

7.14286 points   

QUESTION 8

1. Find the indicated critical z value (use the table given in class)
   
   Find the critical value z _α/2 that corresponds to a 94% confidence level.

7.14286 points   

QUESTION 9

1. Do one of the following, as appropriate: (a) Find the critical value z_α/2, (b) find the critical value t_α/2, (c) state that neither the normal nor the t distribution applies.
   
   90%; n = 10; σ is unknown; population appears to be normally distributed.

   		Neither the normal nor the t distribution applies.
   		tα/2 = 1.812
   		zα/2 = 1.645
   		tα/2 = 1.833

7.14286 points   

QUESTION 10

1. Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p.
   
   A survey of 300 union members in New York State reveals that 112 favor the Republican candidate for governor. Construct the 98% confidence interval for the true population proportion of all New York State union members who favor the Republican candidate.

   		0.304 < p < 0.442
   		0.301 < p < 0.445
   		0.308 < p < 0.438
   		0.316 < p < 0.430

7.14286 points   

QUESTION 11

1. Do one of the following, as appropriate: (a) Find the critical value z_α/2, (b) find the critical value t_α/2, (c) state that neither the normal nor the t distribution applies.
   
   91%; n = 45; σ is known; population appears to be very skewed.

   		Neither the normal nor the t distribution applies.
   		tα/2 = 1.34
   		zα/2 = 1.70
   		tα/2 = 1.645

7.14286 points   

QUESTION 12

1. Use the confidence level and sample data to find the margin of error E. Round your answer to the same number of decimal places as the sample mean unless otherwise noted.
   
   College students' annual earnings in dollars: 99% confidence;      

   		196
   		233
   		258
   		891

7.14286 points   

QUESTION 13

1. Use the given data to find the minimum sample size required to estimate the population proportion.
   
   Margin of error: 0.04; confidence level: 95%; from a prior study,   is estimated by the decimal equivalent of 89%.

   		9
   		236
   		209
   		708

7.14286 points   

QUESTION 14

1. Use the given data to find the minimum sample size required to estimate the population proportion.
   
   Margin of error: 0.005; confidence level: 97%;   and   unknown

   		47,089
   		37,127
   		46,570
   		47,180

Identify the appropriate statistical test for each research description:

1. Single-sample z-test

2. Single-samplet-test

3. Independent samples t-test

4. Related samples t-test (repeated measures or matched samples)

5. Correlation

A researcher investigates whether single people who own pets are generally happier than singles without pets. A group of non-pet owners is compared to pet owners using a mood inventory. The pet owners are matched one to one with the non-pet owners for income, number of close friendships, and general health.

A die is tossed 600 times. H0 is the hypothesis that the proportion of tosses showing aces is binomially distributed with mean 1/6. Find the upper limit of the region for which H0 is accepted at the 1% level of significance in a two sided test.

Remember to SHOW ALL YOUR WORK, and INCLUDE ALL STEPS for hypothesis tests.

1. A sociologist is investigating the relationship between life satisfaction and sociability. For a sample of 16 individuals, the following data is obtained:

SP = 5.14
Sociability: M=10.62 SS=5.12 Life Satisfaction: M=9.53 SS=7.24

a. Compute the correlation between sociability and life satisfaction.
b. What can be said about the nature of the relationship between these two variables? c. Compute the coefficient of determination. What does this value tell you?
d. Conduct a hypothesis test to determine if the correlation is significant.

An investment company knows that the rates of return on its portfolios have a mean of 7.45 percent, with a standard deviation of 3.82 percent. The company selects a sample of 144 portfolios to analyze. Assume the company has tens of thousands of portfolios. (Careful- "percent" is just a unit here!) You do NOT need to check CLT here.

A. What is the probability that the mean of the sample is smaller than 7 percent?

B. What is the probability that the mean of the sample is larger than 7.3 percent?

C. What is the probability that the mean of the sample is between 7.5 and 8.2 percent?

a. We are testing H₀: μ₁ - μ₂ = 0. Our 95% confidence interval is (-27.01,-7.5).
We should expect the t-statistic to be  ---Select--- greater than 2 between 0 and 2 between 0 and -2 less than -2 .
We should expect the p-value to be  ---Select--- less than .05 greater than .05 equal to .05 .
We should  ---Select--- reject fail to reject H₀ and conclude that the group 1 population average is  ---Select--- smaller larger than the group 2 population average.
It is possible that we could be making a  ---Select--- Type I Type II error.

b. We are testing H₀: μ = 15. Our t statistic is 1.25.
We can tell that in our sample, the sample average was  ---Select--- greater less than 15.
We should expect the 95% confidence interval to  ---Select--- include exclude 15.
We should expect the p-value to be  ---Select--- less than .05 greater than .05 equal to .05 .
We should  ---Select--- reject fail to reject H₀.
It is possible that we could be making a  ---Select--- Type I Type II error.

1) You are interested in testing if there is a difference in weight-loss between 3 popular diet types; low calorie, low fat and low carb. Participants are randomly assigned one of the three groups or a control group. The control group will be told that are participating in a weight-loss study, but will not be following any particular diet. This is to study the placebo effect, weight-loss from just participating in the study.
At the start of the study,each participant's weight is recorded in pounds. After 8 weeks they are weighed again, and the weight difference is recorded. A positive value represents a weight loss while a negative value represents a weight gain. Test at 5% significance.

What is the factor variable? Select an answer Diet Type Pounds Lost
What is the response variable? Select an answer Pounds Lost Diet Type
Test Statistic:
P-Value:
Decision Rule: Select an answer Accept the Null Reject the Null Fail to Reject the Null
Did Something Significant Happen? Select an answer Nothing Significant Happened Significance Happened
There Select an answer is is not  to conclude Select an answer The true average weight loss is the same for all diet types. At least one true average weight loss is different between the diet types.

2) Is a statistics class' delivery type a factor in how well students do on the final exam? The table below shows the average percent on final exams from several randomly selected classes that used the different delivery types.

Assume that all distributions are normal, the three population standard deviations are all the same, and the data was collected independently and randomly. Use a level of significance of α=0.1α=0.1.

1. For this study, we should use Select an answer χ²GOF-Test 2-PropZTest TInterval 2-PropZInt 2-SampTTest 1-PropZTest 1-PropZInt T-Test 2-SampTInt χ²-Test ANOVA
2. 
   
3. Your friend Esmeralda helped you with the null and alternative hypotheses...
   H0: μ1=μ2=μ3H0: μ1=μ2=μ3
   H1:H1: At least one of the mean is different from the others.
   
4. The test-statistic for this data = (Please show your answer to 3 decimal places.)
5. 
   
6. The p-value for this sample =  (Please show your answer to 4 decimal places.)
7. 
   
8. The p-value is Select an answer greater than alpha less than (or equal to) alpha  αα
9. 
   
10. Base on this, we should Select an answer reject the null hypothesis fail to reject the null hypothesis accept the null hypothesis  hypothesis
11. 
    
12. As such, the final conclusion is that...
    • There is sufficient evidence to support the claim that course delivery type is a factor in final exam score.
    • There is insufficient evidence to support the claim that course delivery type is a factor in final exam score.

The following three independent random samples are obtained from three normally distributed populations with equal variances. The dependent variable is starting hourly wage, and the groups are the types of position (internship, co-op, work study). Round answers to 4 decimal places.

Use Excel to conduct a single-factor ANOVA to determine if the group means are equal using α=0.02α=0.02.  

Group means:
Internship:   
Co-op:   
Work Study:   

Fill in the summary table for the ANOVA test:

From this table, obtain the necessary statistics for the ANOVA:

ANOVA summary statistics:
Test Statistic =

p-value =

Conclusion: Select an answer The evidence suggests that the average starting hourly wages are different. There is not sufficient evidence to conclude the starting wages are different for the different groups.