If a random sample of 16 homes south of Center Street in Provo has a mean selling price of $145,450 and a standard deviation of $4525, and a random sample of 28 homes north of Center Street has a mean selling price of $148,900 and a standard deviation of $5625, can you conclude that there is a significant difference between the selling price of homes in these two areas of Provo at the 0.05 level? Assume normality. (a) Find t. (Give your answer correct to two decimal places.)

(ii) Find the p-value. (Give your answer correct to four decimal places.)

(b) State the appropriate conclusion.

Reject the null hypothesis, there is not significant evidence of a difference in means.

Reject the null hypothesis, there is significant evidence of a difference in means.

Fail to reject the null hypothesis, there is significant evidence of a difference in means.

Fail to reject the null hypothesis, there is not significant evidence of a difference in means.

11. Determine the p-value given the stated hypothesis and the test statistic value (Z) Ho:μ= 300 H1: μ < 300; z = -2.13 Answer to 4 decimal places.

12. Determine the p-value given the stated hypothesis and test statistic value (Z). Ho: μ = 120 H₁: μ ≠ 120; z = 1.92. Answer to 4 decimal places.

13. Some people claim that the physical demand on dancers are such that dancers tend to be shorter than the typical person. Nationally, adult heights are normally distributed with an average of 64.5 inches. A random sample of 20 adult dancers has an average height of 63.275 inches and a standard deviation of 2.17 inches. Is there evidence that dancers are shorter? Calculate the test-statistic for a test

Ho: μ = 64.5 versus Ha: μ < 64.5

Compute the test statistic for this test. (round your answer to 2 decimal places)

14. The mean caffeine content per cup of regular coffee served at a certain coffee shop is supposed to be 100mg. A test is made of Ho: μ=100 versus Ha: μ≠100. A sample of 35 cups does not provide enough evidence that the mean caffeine content is different from 100mg. Which type of error is possible in this situation?

	A.	Type 2 error
	B.	Type 1 error
	C.	Impossible to tell
	D.	Either type is possible

A survey was conducted in which 125 families were asked how many cats lived in their households. The results are shown below. ​a) What is the probability that a randomly selected family has one​ cat? ​b) What is the probability that a randomly selected family has more than one​ cat? ​c) What is the probability that a randomly selected family has​ cats? ​d) Is this an example of​ classical, empirical, or subjective​ probability? *Number of Cats-Number of Households* 0-71 1-33 2-16 3-4 4-1 Total-125 ​a) The probability that a randomly selected household has one cat is nothing. ​(Simplify your​ answer.) ​b) The probability that a randomly selected family has more than one cat is nothing. ​(Simplify your​ answer.) ​c) The probability that a randomly selected family has cats is nothing. ​(Simplify your​ answer.) ​d) Is this an example of​ classical, empirical, or subjective​ probability? Empirical probability Classical probability Subjective probability Thank you in advance!!

11.You measure the weight of 37 turtles, and find they have a mean weight of 52 ounces. Assume the population standard deviation is 10.9 ounces. Based on this, what is the maximal margin of error associated with a 92% confidence interval for the true population mean turtle weight.
Give your answer as a decimal, to two places

12.In a survey, 18 people were asked how much they spent on their child's last birthday gift. The results were roughly shaped as a normal curve with a mean of $45 and standard deviation of $8. Find the margin of error at a 80% confidence level.

13. In a survey, 20 people were asked how much they spent on their child's last birthday gift. The results were roughly shaped as a normal curve with a mean of $34 and standard deviation of $12. Construct a confidence interval at a 80% confidence level.

Give your answers to one decimal place.

.................. ± ..............

Age	Mileage
6	53808
7	82838
11	115903
6	54903
8	77564
10	95911
4	40686
12	126675
15	167636
14	128798
10	96589
5	35049

A used car dealer wants to develop a regression equation that determines mileage as a function of the age of a car in years. He collects the data shown below for the 12 cars he has on his lot.

a) What is the slope of the regression equation? Give your answer to two decimal places.  
b) What is the value of the correlation coefficient? Give your answer to two decimal places.  
c) A 4 year old car is delivered to his lot with 160000 miles. Manually enter these values in the data table above and rerun the regression analysis. What is the value of the slope? Give your answer to two decimal places.  
d) Including the additional car, what is the value of the correlation coefficient? Give your answer to two decimal places.  
e) Did the additional car strengthen or weaken the linear relationship between age and mileage?

It strengthened the linear relationship.

It weakened the linear relationship.

   Can not be determined.

ID of Respondent	# of Friends who Bully	Respondent was a Bully Victim (0 = No; 1 = Yes)	Gender (0 = Female; 1 = Male)	# of Times Respondent Bullied Others
1	2	1	1	5
2	4	1	0	2
3	3	0	1	8
4	2	0	0	4
5	6	1	1	6
6	3	0	0	2
7	7	1	1	7
8	4	0	0	0
9	2	1	1	1
10	7	1	1	8

1.         What is the proportion of males who bullied others?  What is the proportion of females who bullied others?  Which gender (male or female) possessed a deeper involvement in bullying others?

Kroger is in the process of designing a new store to be located in a plaza under development in Mason. They intend to use their Symmes Township store as a model, but they are concerned that the customer base in Mason might have different needs and expectations. One area of concern is in Service Meats. Grocery shoppers in Symmes Township expect a Service Meat counter, and the department has been quite profitable. Kroger would like to know if the expectation of having a Service Meat counter will be the same in Mason as it is in Symmes Township. They survey residents of both areas, and among the questions is, "Do you buy meat from the Service Meat counter on a regular (weekly) basis?" In city a, 505 out of 780 respondents said YES. In city b, 325 out of 620 respondents said YES. Using a = .05, test the claim that the percentage of grocery shoppers who use Service Meats is the same in these two areas of the city.

Suppose you are interested in buying a new Toyota Corolla. You are standing on the sales lot looking at a model with different options. The list price is on the vehicle. As a salesperson approaches, you wonder what the dealer invoice price is for this model with its options. The following data are based on a random selection of Toyota Corollas of different models and options. Let y be the dealer invoice (in thousands of dollars) for the given vehicle.

x	12.9	13.0	12.8	13.6	13.4	14.2
y	11.6	11.9	11.5	12.2	12.0	12.8

(a) Verify that Σx = 79.9, Σy = 72, Σx² = 1065.41, Σy² = 865.1, Σxy = 960.02, and r ≈ 0.980.

Σx	=
Σy	=
Σx2	=
Σy2	=
Σxy	=
r	=

(b) Use a 1% level of significance to test the claim that ρ > 0. (Use 2 decimal places.)

t	=
critical t	=

Conclusion

Reject the null hypothesis, there is sufficient evidence that ρ > 0.

Reject the null hypothesis, there is insufficient evidence that ρ > 0.    

Fail to reject the null hypothesis, there is insufficient evidence that ρ > 0.

Fail to reject the null hypothesis, there is sufficient evidence that ρ > 0.

(c) Verify that S_e ≈ 0.1039, a ≈ 0.464, and b ≈ 0.866.

Se	=
a	=
b	=

(d) Find the predicted dealer invoice when the list price is x = 13.6 (thousand dollars). (Use 2 decimal places.)


(e) Find a 90% confidence interval for y when x = 13.6 (thousand dollars). (Use 2 decimal places.)

lower limit	=
upper limit	=

(f) Use a 1% level of significance to test the claim that β > 0. (Use 2 decimal places.)

t	=
critical t	=

Conclusion

Reject the null hypothesis, there is sufficient evidence that β > 0.

Reject the null hypothesis, there is insufficient evidence that β > 0.    

Fail to reject the null hypothesis, there is insufficient evidence that β > 0.

Fail to reject the null hypothesis, there is sufficient evidence that β > 0.

(g) Find a 90% confidence interval for β and interpret its meaning. (Use 2 decimal places.)

lower limit	=
upper limit	=

Interpretation

For every $1,000 increase in list price, the dealer price increases by an amount that falls within the confidence interval.

For every $1,000 increase in list price, the dealer price increases by an amount that falls outside the confidence interval.    

For every $1,000 increase in list price, the dealer price decreases by an amount that falls within the confidence interval.

For every $1,000 increase in list price, the dealer price decreases by an amount that falls outside the confidence interval.

ID of Respondent	# of Friends who Bully	Respondent was a Bully Victim (0 = No; 1 = Yes)	Gender (0 = Female; 1 = Male)	# of Times Respondent Bullied Others
1	2	1	1	5
2	4	1	0	2
3	3	0	1	8
4	2	0	0	4
5	6	1	1	6
6	3	0	0	2
7	7	1	1	7
8	4	0	0	0
9	2	1	1	1
10	7	1	1	8

If you were to draw one individual from this sample of 10 individuals, what would do characteristics do you believe would be the most likely to be drawn (1 = Bully Victim and Bully Offender; 2 = Bully Victim but not Bully Offender; 3 = Bully Offender but not Bully Victim; or 4 = Non Bully Victim and Non Bully Offender)?  Why?

The following table shows ceremonial ranking and type of pottery sherd for a random sample of 434 sherds at an archaeological location.

Ceremonial Ranking	Cooking Jar Sherds	Decorated Jar Sherds (Noncooking)	Row Total
A	91	44	135
B	88	57	145
C	80	74	154
Column Total	259	175	434

Use a chi-square test to determine if ceremonial ranking and pottery type are independent at the 0.05 level of significance.

(a) What is the level of significance?
.05

State the null and alternate hypotheses.

H₀: Ceremonial ranking and pottery type are not independent.
H₁: Ceremonial ranking and pottery type are not independent.

H₀: Ceremonial ranking and pottery type are independent.
H₁: Ceremonial ranking and pottery type are not independent.

   H₀: Ceremonial ranking and pottery type are not independent.
H₁: Ceremonial ranking and pottery type are independent.

H₀: Ceremonial ranking and pottery type are independent.
H₁: Ceremonial ranking and pottery type are independent.

(b) Find the value of the chi-square statistic for the sample. (Round the expected frequencies to at least three decimal places. Round the test statistic to three decimal places.)


Are all the expected frequencies greater than 5?

What sampling distribution will you use?

chi-square

normal   

Student's t

binomial

uniform

What are the degrees of freedom?


(c) Find or estimate the P-value of the sample test statistic. (Round your answer to three decimal places.)

p-value > 0.1000

.050 < p-value < 0.100   

0.025 < p-value < 0.0500

.010 < p-value < 0.0250

.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 of independence?

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 sufficient evidence to conclude that ceremonial ranking and pottery type are not independent.At the 5% level of significance, there is insufficient evidence to conclude that ceremonial ranking and pottery type are not independent.  

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 51 women in rural Quebec gave a sample variance s² = 2.4. 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?
.05

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 exponential population distribution.    

We assume a binomial population distribution.

We assume a uniform 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 < 0.0500

.010 < P-value < 0.0250

.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.)

lower limit
upper limit

Interpret the results in the context of the application.

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

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

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

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

Suppose the probability mass function of a random variable X is given by

??x−1?pr(1−p)x−r, ifx=r,r+1,r+2,... f(x) = r−1

0, otherwise

If this is the case then we say X is distributed as a Negative Binomial Random Variable with parameters r and p and we write X ∼ NegBin(r, p) (a) If we set r = 1, what distribution do we get? (b) Explain what this random variable models and justify the formula. (Hint: See Section 4.8.2 in Ross.) Math 241 Quiz 3 - Page 2 of 2 August 2019 (c) For this random variable, what is E[X] and Var[X]? There is no need to prove your answer or show any work for this part. (Hint: See Section 4.8.2 in Ross.) (d) In tossing a fair die repeatedly (and independently on successive tosses), find the proba- bility of getting the third “1” on the xth toss. (Hint: Let X denote the number of tosses required until we get our third “1”, or equivalently our third success. Then X is what kind of random variable?) (e) In tossing a fair die repeatedly (and independently on successive tosses), find the proba- bility of getting the third “1” on the fifth toss. (f) What is the average number of trials it will take to get our third “1”? (Hint: Use the results of part (c) for your solution).

Problem 11-13

The Martin-Beck Company operates a plant in St. Louis with an annual capacity of 30,000 units. Product is shipped to regional distribution centers located in Boston, Atlanta, and Houston. Because of an anticipated increase in demand, Martin-Beck plans to increase capacity by constructing a new plant in one or more of the following cities: Detroit, Toledo, Denver, or Kansas City. The estimated annual fixed cost and the annual capacity for the four proposed plants are as follows:

Proposed Plant	Annual Fixed Cost	Annual Capacity
Detroit	$175,000	10,000
Toledo	$300,000	20,000
Denver	$375,000	30,000
Kansas City	$500,000	40,000

The company's long-range planning group developed forecasts of the anticipated annual demand at the distribution centers as follows:

Distribution Center	Annual Demand
Boston	30,000
Atlanta	20,000
Houston	20,000

The shipping cost per unit from each plant to each distribution center is shown in table below.

A network representation of the potential Martin-Beck supply chain is shown in figure below.

Each potential plant location is shown; capacities and demands are shown in thousands of units. This network representation is for a transportation problem with a plant at St. Louis and at all four proposed sites. However, the decision has not yet been made as to which new plant or plants will be constructed.

1. Formulate a model that could be used for choosing the best plant locations and for determining how much to ship from each plant to each distribution center. There is a policy restriction that a plant must be located either in Detroit or in Toledo, but not both. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300)

Let
	y1 = 1 if a plant is constructed in Detroit; 0 if not
	y2 = 1 if a plant is constructed in Toledo; 0 if not
	y3 = 1 if a plant is constructed in Denver; 0 if not
	y4 = 1 if a plant is constructed in Kansas City; 0 if not
	xij = the units shipped in thousands from plant i to distribution center j
	i= 1,2,3,4,5, and j = 1,2,3

Min

x11

+

x12

+

x13

+

x21

+

x22

+

x23

+

x31

+

x32

+

x33

+

x41

+

x42

+

x43

+

x51

+

x52

+

x53

+

y1

+

y2

+

y3

+

y4

1. Formulate a model that could be used for choosing the best plant locations and for determining how much to ship from each plant to each distribution center. There is a policy restriction that no more than two plants can be located in Denver, Kansas City, and St. Louis. For those boxes in which you must enter subtractive or negative numbers use a minus sign. (Example: -300)

Let
	y1 = 1 if a plant is constructed in Detroit; 0 if not
	y2 = 1 if a plant is constructed in Toledo; 0 if not
	y3 = 1 if a plant is constructed in Denver; 0 if not
	y4 = 1 if a plant is constructed in Kansas City; 0 if not
	xij = the units shipped in thousands from plant i to distribution center j
	i= 1,2,3,4,5, and j = 1,2,3

Min

x11

+

x12

+

x13

+

x21

+

x22

+

x23

+

x31

+

x32

+

x33

+

x41

+

x42

+

x43

+

x51

+

x52

+

x53

+

y1

+

y2

+

y3

+

y4

Please show how to solve parts a and b using Excel

2. Make a data frame consisting of 20 and 10 columns. Each column j should consist of 20 values from a normal distribution with mean (i-1) and standard deviation 0.5j. For example, the third column should be normal(mean=2, sd=1.5). Using this data frame, do each of the following (using code, of course):
a. Find the mean and standard deviation for each column.
b. Write code that counts the number of columns for which the sample mean and sample standard deviation are within 20% of the values used to generate the data.
c. Write code that writes the columns from part b to a new data frame.
d. For each value in the new data frame, subtract its column mean and divide by the column standard deviation.

Solution using r and python

The age distribution of the Canadian population and the age distribution of a random sample of 455 residents in the Indian community of a village are shown below.

Age (years)	Percent of Canadian Population	Observed Number in the Village
Under 5	7.2%	44
5 to 14	13.6%	75
15 to 64	67.1%	289
65 and older	12.1%	47

Use a 5% level of significance to test the claim that the age distribution of the general Canadian population fits the age distribution of the residents of Red Lake Village.

(a) What is the level of significance?
.05

State the null and alternate hypotheses.

H₀: The distributions are the same.
H₁: The distributions are the same.

H₀: The distributions are different.
H₁: The distributions are the same.    

H₀: The distributions are the same.
H₁: The distributions are different.

H₀: The distributions are different.
H₁: The distributions are different.

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


Are all the expected frequencies greater than 5?

  

What sampling distribution will you use?

Student's t

uniform

   normal

binomial

chi-square

What are the degrees of freedom?


(c) Estimate the P-value of the sample test statistic.

P-value > 0.1000

.050 < P-value < 0.100  

  0.025 < P-value < 0.0500

.010 < P-value < 0.0250

.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 that the population fits the specified distribution of categories?

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, the evidence is insufficient to conclude that the village population does not fit the general Canadian population.

At the 5% level of significance, the evidence is sufficient to conclude that the village population does not fit the general Canadian population.