Assume a normal distribution of the form N(100, 100), answer the following questions:

What proportion of the distribution falls within 1 standard deviation of the mean (e.g., within -1 and 1 standard deviation)?

What is the probability of a single draw from that distribution has a value greater than 115?

What is the range that captures the middle 95% of the population distribution?

If I randomly sample 10 observations from this distribution and calculate a mean, what is the probability that this mean is greater than 106?

If we center the sampling distribution on 100, then what is the range that will capture 95% of the means calculated from a sample of 20 observations?

The yield of a chemical process is being studied. The two most important variables are thought to be the pressure and the temperature. Three levels of each factor are selected, and a factorial experiment with two replicates is performed. The yield data are as follows:

	Pressure (psig)
Temperature (ºC)	200	215	230
150	90.4	90.7	90.2
	90.2	90.6	90.4
160	90.1	90.5	89.9
	90.3	90.6	90.1
170	90.5	90.8	90.4
	90.7	90.9	90.1

1. Use the LSD test to determine which levels of the pressure factor are significantly different.
2. Use the LSD test to determine which levels of the temperature factor are significantly different.
3. Suppose that we wish to reject the null hypothesis with a high probability if the difference in the true mean yield at any two pressures is as great as 0.5. If a reasonable prior estimate of the standard deviation of yield is 0.1, how many replicates should be run?

Test if the population mean price for clarity “VS1” is different than that for clarity “VVS1 or VVS2”.

Please answer with R programming code

Clarity	Price
VS2	1302
VS1	1510
VVS1	1510
VS1	1260
VS1	1641
VS1	1555
VS1	1427
VVS2	1427
VS2	1126
VS1	1126
VS1	1468
VS2	1202
VS2	1327
VS2	1098
VS1	1693
VS1	1551
VS1	1410
VS2	1269
VS1	1316
VS2	1222
VS1	1738
VS1	1593
VS1	1447
VS2	1255
VS1	1635
VVS2	1485
VS2	1420
VS1	1420
VS1	1911
VS1	1525
VS1	1956
VVS2	1747
VS1	1572
VVS2	2942
VVS2	2532
VS1	3501
VS1	3501
VVS2	3501
VS1	3293
VS1	3016
VVS2	3567
VS1	3205
VS2	3490
VS1	3635
VVS2	3635
VS1	3418
VS1	3921
VVS2	3701
VS1	3480
VVS2	3407
VS1	3767
VVS1	4066
VVS2	4138
VS1	3605
VVS2	3529
VS1	3667
VVS2	2892
VVS2	3651
VVS2	3773
VS1	4291
VVS1	5845
VVS2	4401
VVS1	4759
VVS1	4300
VS1	5510
VS1	5122
VVS2	5122
VS2	3861
VVS2	5881
VS1	5586
VS2	5193
VVS2	5193
VS2	5263
VVS2	5441
VS2	4948
VS2	5705
VS2	6805
VVS2	6882
VS1	6709
VVS2	6682
VS1	3501
VVS1	3432
VVS1	3851
IF	3605
VS1	3900
VVS1	3415
IF	4291
IF	6512
VS1	5800
VVS1	6285

Question 15.

What are the three required assumptions for the appropriate use of the independent groups t-test? What are the three required assumptions for the appropriate use of the dependent groups t-test?Can you use these tests when you have three groups? What test do we use instead? Can the dependent variable be nominal? What should the nature of the dependent variable be?

  

  

A team of health research wants to investigate whether having regular lunch break improves working adults’ sleep. A group of 85 adults with a full-time job were recruited in the study, they reported average hours of sleep in the past week, committed to having a 1-hour, out-of-office, work-free lunch break each day for 3 months, and, at the end of the study, reported average hours of sleep in the past week. The mean difference in hours of sleep before and after the study was 3 with a standard deviation of 0.9. Which one of the following statements is INCORRECT? (Set alpha level at 0.05.)

The decision should be to reject the null hypothesis.

The null hypothesis is that hours of sleep remain the same before and after the study.

The obtained test statistic is 3.073.

The degrees of freedom is 84.

We have 95 students in a class. Their abilities/eagerness are uniform randomly distributed on a scale between 1 and 4; and at the end of the class they will be judged right and they will receive a grade corresponding to their ability/eagerness (corresponding to their performance). What is the probability that the class average will be between 2.8 and 4? How would this number change (if it does) for 120 students?

(PLEASE, SINCE THE VERY BEGINNING, ALL THE ONE BY ONE STEPS NEED TO BE MENTIONED IN YOUR CALCULATION) A manufacturer of cell phones guarantees that his cell phones will last, on average, 3 years with a standard deviation of 1 year. If 5 of those cell phones are found to have lifetimes of 1.9, 2.4, 3.0, 3.5 and 4.2 years, can the manufacturer still be convinced that his cell phones have a standard deviation of 1 year? Test at a 0.05 level of confidence. Thank you in advance for your help!

Eyeglassomatic manufactures eyeglasses for different retailers. They test to see how many defective lenses they made the time period of January 1 to March 31. Table #11.2.4 gives the defect and the number of defects.

Table #11.2.4: Number of Defective Lenses

Defect type	Number of defects
Scratch	5865
Right shaped - small	4613
Flaked	1992
Wrong axis	1838
Chamfer wrong	1596
Crazing, cracks	1546
Wrong shape	1485
Wrong PD	1398
Spots and bubbles	1371
Wrong height	1130
Right shape - big	1105
Lost in lab	976
Spots/bubble - interim	976

Do the data support the notion that each defect type occurs in the same proportion? Test at the 10% level.

The types of raw materials used to construct stone tools found at an archaeological site are shown below. A random sample of 1486 stone tools were obtained from a current excavation site.

Raw Material	Regional Percent of Stone Tools	Observed Number of Tools as Current excavation Site
Basalt	61.3%	905
Obsidian	10.6%	163
Welded Tuff	11.4%	165
Pedernal chert	13.1%	201
Other	3.6%	52

Use a 1% level of significance to test the claim that the regional distribution of raw materials fits the distribution at the current excavation site.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: The distributions are the same.
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.

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

(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?

Yes

No    

What sampling distribution will you use?

normal

Student's t    

binomial

uniform

chi-square

What are the degrees of freedom?


(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 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 0.01 level of significance, the evidence is sufficient to conclude that the regional distribution of raw materials does not fit the distribution at the current excavation site.

At the 0.01 level of significance, the evidence is insufficient to conclude that the regional distribution of raw materials does not fit the distribution at the current excavation site.

Use the data in the file andy.dta consisting of data on hamburger franchises in 75 cities from Big Andy's Burger Barn.

Set up the model

ln(Si)=b1 + b2ln(Ai) + ei,

where

Si = Monthly sales revenue ($1000s) for the i-th firm

Ai = Expenditure on advertising ($1000s) for the i-th firm

(a) Interpret the estimates of slope and intercept.

(b) How well did the model fit to the data? Use any tests and measures presented in class.

(c) Perform any test for heteroscedasticity in your data.

sales	price	advert
73.2	5.69	1.3
71.8	6.49	2.9
62.4	5.63	0.8
67.4	6.22	0.7
89.3	5.02	1.5
70.3	6.41	1.3
73.2	5.85	1.8
86.1	5.41	2.4
81	6.24	0.7
76.4	6.2	3
76.6	5.48	2.8
82.2	6.14	2.7
82.1	5.37	2.8
68.6	6.45	2.8
76.5	5.35	2.3
80.3	5.22	1.7
70.7	5.89	1.5
75	5.21	0.8
73.7	6	2.9
71.2	6.37	0.5
84.7	5.33	2.1
73.6	5.23	0.8
73.7	5.88	1.1
78.1	6.24	1.9
75.7	5.59	2.1
74.4	6.22	1.3
68.7	6.41	1.1
83.9	4.96	1.1
86.1	4.83	2.9
73.7	6.35	1.4
75.7	6.47	2.5
78.8	5.69	3
73.7	5.56	1
80.2	6.41	3.1
69.9	5.54	0.5
69.1	6.47	2.7
83.8	4.94	0.9
84.3	6.16	1.5
66	5.93	2.8
84.3	5.2	2.3
79.5	5.62	1.2
80.2	5.28	3.1
67.6	5.46	1
86.5	5.11	2.5
87.6	5.04	2.1
84.2	5.08	2.8
75.2	5.86	3.1
84.7	4.89	3.1
73.7	5.68	0.9
81.2	5.83	1.8
69	6.33	3.1
69.7	6.47	1.9
78.1	5.7	0.7
88	5.22	1.6
80.4	5.05	2.9
79.7	5.76	2.3
73.2	6.25	1.7
85.9	5.34	1.8
83.3	4.98	0.6
73.6	6.39	3.1
79.2	6.22	1.2
88.1	5.1	2.1
64.5	6.49	0.5
84.1	4.86	2.9
91.2	5.1	1.6
71.8	5.98	1.5
80.6	5.02	2
73.1	5.08	1.3
81	5.23	1.1
73.7	6.02	2.2
82.2	5.73	1.7
74.2	5.11	0.7
75.4	5.71	0.7
81.3	5.45	2
75	6.05	2.2

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 = 2.7. 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 binomial population distribution.We assume a normal population distribution.    We assume a uniform 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.)

lower limit
upper limit

Interpret the results in the context of the application.

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.

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

1. A recent poll was conducted by the Gallup organization between April 2nd and April 8th, 2018. A total of 785 Facebook users living in the U.S. were selected using random digit dialing and were interviewed over the phone (either landline or cell phone). Respondents were asked the following question: “How concerned are you about invasion of privacy when using Facebook? Very concerned, somewhat concerned, not too concerned or not concerned.” 43% of those selected said “very concerned”. Suppose we want to study the proportion of all Facebook users who are “very concerned” about the invasion of privacy.

a) Define the parameter of interest in the context of the problem (include the symbol used to denote it).

b) What is the statistic in this problem (include the symbol used to denote it)?

c) Check the assumptions necessary to construct a normal-based confidence interval for the parameter of interest by verifying the appropriate conditions. Actually show that you checked these in the context of the problem.

d) Find the margin of error if we want 95% confidence in our estimate of the parameter of interest.

e) Find the 95% confidence interval for the parameter of interest.

f) In the context of the problem, give a conclusion based on the confidence interval you found in part e. Your sentence should start with the words “We are 95% confident that……”

Hi,

Im doing my final project for quanatative analysis class. I was tasked to create a regression analysis on Does the number of probowl player have bearing on becoming all pro. I ran the regression for the following data and got this output from Megastat but I'm not sure what it tells me.

Regression Analysis
	r²	0.006	n	239
	r	0.076	k	1
	Std. Error	0.494	Dep. Var.	All Pro
ANOVA table
Source	SS	df	MS	F	p-value
Regression	0.3397	1	0.3397	1.39	.2392
Residual	57.8193	237	0.2440
Total	58.1590	238
Regression output					confidence interval
variables	coefficients	std. error	t (df=237)	p-value	95% lower	95% upper
Intercept	-0.1681	0.4981	-0.337	.7361	-1.1493	0.8131
Pro Bowl	0.5840	0.4950	1.180	.2392	-0.3911	1.5591

Sum of Pro Bowl Count	Sum of All Pro Count
240	101

The ideal (daytime) noise-level for hospitals is 45 decibels with a standard deviation of 9 db; which is to say, this may not be true. A simple random sample of 80 hospitals at a moment during the day gives a mean noise level of 47 db. Assume that the standard deviation of noise level for all hospitals is really 9 db. All answers to two places after the decimal.
(a) A 99% confidence interval for the actual mean noise level in hospitals is ( ___db, ___ db).
(b) We can be 90% confident that the actual mean noise level in hospitals is  db with a margin of error of ___ db.
(c) Unless our sample (of 81 hospitals) is among the most unusual 2% of samples, the actual mean noise level in hospitals is between ___ db and ___ db.
(d) A 99.9% confidence interval for the actual mean noise level in hospitals is (___ db, ___ db).
(e) Assuming our sample of hospitals is among the most typical half of such samples, the actual mean noise level in hospitals is between ___ db and ___db.
(f) We are 95% confident that the actual mean noise level in hospitals is  db, with a margin of error of ___db.
(g) How many hospitals must we examine to have 95% confidence that we have the margin of error to within 0.25 db?
(h) How many hospitals must we examine to have 99.9% confidence that we have the margin of error to within 0.25 db?

According to a genetic model, the distribution of fur color of the second generation Havana rabbit (Oryctolagus cuniculus) should be 1:2:1, black:gray:brown. In a sample of second generation rabbits, there are 10 black, 27 gray, and 16 brown rabbits. Assuming random sample and independent observations, does this sample of rabbits suggest that the actual fur color distribution differs from the genetic model? Include all steps for full credit.