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.

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.

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.)

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.

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.

Assuming the sample represents the population very well, that is, the population has approximately the same mean and same standard deviation.

a) 68% of the population fall between ___ inches and ___ inches

b) 95% of the population fall between ___ inches and ___ inches

c) 99.7% of the population fall between ___ inches and ___ inches

Please answer (F)

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 ≠ 5.1H_o: σ² = 5.1; H₁: σ² < 5.1    H_o: σ² < 5.1; H₁: σ² = 5.1H_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 > 0.1000.050 < P-value < 0.100    0.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.010P-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.)

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.

The reading speed of second grade students in a large city is approximately​ normal, with a mean of 90 words per minute​ (wpm) and a standard deviation of 10 wpm. Complete parts​ (a) through​ (f).

​(a) What is the probability a randomly selected student in the city will read more than 95 words per​ minute?

​(b) What is the probability that a random sample of 13 second grade students from the city results in a mean reading rate of more than 95 words per​ minute? then interpret this probability.

(c) What is the probability that a random sample of 26 second grade students from the city results in a mean reading rate of more than 95 words per​ minute? then interpret.

(d) What effect does increasing the sample size have on the​ probability? Provide an explanation for this result.

(e) A teacher instituted a new reading program at school. After 10 weeks in the​ program, it was found that the mean reading speed of a random sample of 20 second grade students was 92.3 wpm. What might you conclude based on this​ result? Select the correct choice below and fill in the answer boxes within your choice.

(f) There is a​ 5% chance that the mean reading speed of a random sample of 24 second grade students will exceed what​ value?

Yellow stains on white clothing are stubborn discolorations believed to result from oxidated body oils that have reacted chemically with clothing’s cellulose fibers (as if it were a dye). In as little as one week, there reaction can permanently discolor with clothing with an ugly yellow stain and at this point the most potent laundry product will usually be of little value in removing them. Suppose researchers at the Department of Textiles and Apparel at Cornell University, experimenting with new experimental formulations to remove stains, washed six discolored strips in each of three experimental formulations, and the results as follows Coloring rating (after washing) 0 = Dark Yellow, the initial shade 20 = Pure White-All washed strips were matched to a color chart, a series of increasingly lighter shade, and rated from 0 to 20. Formulation X Y Z 10 8 11 9 10 12 10 12 14 12 9 11 8 10 10 11 8 14 Three Different Formulations were tested a) Calculate by hand the average color rating and variance for each group b) Assuming valid random samples, test at 5% level significance whether differences among the three-sample means can be attributed to chance fluctuation or, indeed, to actual differences in the stain-eliminating ability of the three formulations. ( hint: find the F-ratio) c) Briefly discuss the implications of these results. Need to solve by hand. Please help!

A research council wants to estimate the mean length of time (in minutes) that the average U.S. adult spends watching television using digital video recorders (DVR’s) each day. To determine the estimate, the research council takes random samples of 35 U.S. adults and obtains the following times in minutes.

From past studies, the research council has found that the standard deviation time is 4.3 minutes and that the population of times is normally distributed.

Construct a 90% confidence interval for the population mean.

Construct a 99% confidence interval for the population mean.

Interpret the results and compare the widths of the confidence intervals.

Test the claim that the mean time spent watching DVR’s is 20 minutes each day using a significance level of 0.05.

You may use Stat Disk, TI-84 calculator, or CrunchIt to find the confidence intervals. Be sure to show your results in your post.

(3) A random sample of 51 fatal crashes in 2009 in which the driver had a positive blood alcohol concentration (BAC) from the National Highway Traffic Safety Administration results in a mean BAC of 0.167 grams per deciliter (g/dL) with a standard deviation of 0.010 g/dL.

(a) What is a point estimate for the mean BAC of all fatal crashes with a positive BAC?

(b) In 2009, there were approximately 25,000 fatal crashes in which the driver had a positive BAC. Explain why this, along with the fact that the data were obtained using a simple random sample, satisfies the requirements for constructing a confidence interval.

(c) Determine and interpret a 98% confidence interval for the mean BAC in fatal crashes in which the driver had a positive BAC. Do not use the T-Interval or Z-Interval features of your calculator.

8. Given the following data: Number of patients = 3,293 Number of patients who had a positive test result and had the disease = 2,184 Number of patients who had a negative test, and did not have the disease = 997 Number of patients who had a positive test result, but did not have the disease = 55 Number of patients who had a negative test result, but who had the disease = 57 a. Create a complete and fully labeled 2x2 table (10) b. Calculate the positive predictive value (5) c. Calculate the negative predictive value (5) d. Calculate the likelihood ratio for a positive test result (5) e. Calculate the likelihood ratio for a negative test result (5)

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Jan Northcutt, owner of Northcutt Bikes, started business in 1995. She notices the quality of bikes she purchased for sale in her bike shop declining while the prices went up. She also found it more difficult to obtain the features she wanted on ordered bikes without waiting for months. Her frustration turned to a determination to build her own bikes to her particular customer specifications.

She began by buying all the necessary parts (frames, seats, tires, etc.) and assembling them in a rented garage using two helpers. As the word spread about her shop’s responsiveness to options, delivery, and quality, however, the individual customer base grew to include other bike shops in the area. As her business grew and demanded more of her attention, she soon found it necessary to sell the bike shop itself and concentrate on the production of bikes from a fairly large leased factory space.

As the business continued to grow, she backward integrated more and more processes into her operation, so that now she purchases less than 50% of the component value of the manufactured bikes. This not only improves her control of production quality but also helps her control the costs of production and makes the final product more cost attractive to her customers.

The Current Situation

Jan considers herself a hands-on manager and has typically used her intuition and her knowledge of the market to anticipate production needs. Since one of her founding principles was rapid and reliable delivery to customer specification, she felt she needed to begin production of the basic parts for each particular style of bike well in advance of demand. In that way she could have the basic frame, wheels, and standard accessories started in production prior to the recognition of actual demand, leaving only the optional add-ons to assemble once the order came in. Her turnaround time for an order of less than half the industry average is considered a major strategic advantage, and she feels it is vital for her to maintain or even improve on response time if she is to maintain her successful operation.

As the customer base have grown, however, the number of customers Jan knows personally has shrunk significantly as a percentage of the total customer base for Northcutt Bikes, and many of these new customers are expecting or even demanding very short response times, as that is what attracted them to Northcutt Bikes in the first place. This condition, in addition to the volatility of overall demand, has put a strain on capacity planning. She finds that at times there is a lot of idle time (adding significantly to costs), whereas at other times the demand exceeds capacity and hurts customer response time. The production facility has therefore turned to trying to project demand for certain models, and actually building a finished goods inventory of those models. This has not proven to be too satisfactory, as it has actually hurt costs and some response times. Reasons include the following:

- The finished goods inventory is often not the “right” inventory, meaning shortages for some goods and excessive inventory of others. This condition both hurts responsiveness and increases inventory costs.

- Often, to help maintain responsiveness, inventory is withdrawn from finished goods and reworked, adding to product cost.

- Reworking inventory uses valuable capacity for other customer orders, again resulting in poorer response times and/or increased costs due to expediting. Existing production orders and rework orders are both competing for vital equipment and resources during times of high demand, and scheduling has become a nightmare.

The inventory problem has grown to the point that additional storage space is needed, and that is a cost that Jan would like to avoid if possible.

Another problem that Jan faces is the volatility of demand for bikes. Since she is worried about unproductive idle time and yet does not wish to lay off her workers during times of low demand, she has allowed them to continue to work steadily and build finished goods. This makes the problem of building the “right” finished goods even more important, especially given the tight availability of storage space.

Past Demand

The following shows the monthly demand for one major product line: the standard 26-inch 10-speed street bike. Although it is only one of Jan’s products, it is representative of most of the major product lines currently being produced by Northcutt Bikes. If Jan can find a way to sue this data to more constructively understand her demand, she feels she can probably use the same methodologies to project demand for other major product families. Such knowledge can allow her, she feels, to plan more effectively and continue to be responsive while still controlling costs.

1. Plot the data and describe what you see. What does it mean and how would you use the information from the plot to help you develop a forecast?

2. Use at least two different methodologies to develop as accurate a forecast as possible for the demand. Use each of those methods to project the next four months demand.

3. Which method from question 2 is “better”? How do you know that?