One of the products produced by Branco Food Company is All-Bran Cereal, which competes with three other brands of similar all-bran cereals. The company's research office wants to investigate if the percentage of people who consume all-bran cereal is the same for each of these four brands. Let us denote the four brands of cereal by A,B,C, and D. A sample of 900 persons who consume all-bran cereal was taken, and they were asked which brand they most often consume. Of the respondents, 226said they usually consume Brand A, 232 consume Brand B, 233 consume Brand C, and 209 consume Brand D. Does the sample provide enough evidence to reject the null hypothesis that the percentage of people who consume all-bran cereal is the same for all four brands? Use α=0.05.

Assume that a sample is used to estimate a population mean μ . Find the margin of error M.E. that corresponds to a sample of size 18 with a mean of 47.4 and a standard deviation of 16.9 at a confidence level of 90%. Report ME accurate to one decimal place because the sample statistics are presented with this accuracy. M.E. _________

he effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 32.7 for a sample of size 288 and standard deviation 11.5.

Estimate how much the drug will lower a typical patient's systolic blood pressure (using a 80% confidence level).

Enter your answer as a tri-linear inequality accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).

______< μ< ______

SAT scores are distributed with a mean of 1,500 and a standard deviation of 300. You are interested in estimating the average SAT score of first year students at your college. If you would like to limit the margin of error of your 95% confidence interval to 25 points, how many students should you sample?

Make sure to give a whole number answer.

Let X ∈{1,2} and Y ∈{3,4} be independent random variables with PMF-s: fX(1) = 1 2 fX(2) = 1 2 fY (3) = 1 3 fY (4) = 2 3 Answer the following questions (a) Write down the joint PMF

(b) Calculate P(X + Y 6 5) and P(Y −X > 2)

(c) Calculate E(XY ), E(X2Y ), E(︁X2+1 Y−2)︁

(d) Calculate the Cov(X,Y ), Cov(1−X,3Y + 2) and Var(2X −Y ) (

e*) Calculate Cov(XY,X), Cov(XY,X + Y ) and Var(︀X Y)︀

Suppose that a category of world class runners are known to run a marathon (26 miles) in an average of 141 minutes with a standard deviation of 12 minutes. Consider 49 of the races. Let X = the average of the 49 races. Part (a) Give the distribution of X. (Round your standard deviation to two decimal places.) X ~ , Part (b) Find the probability that the runner will average between 138 and 142 minutes in these 49 marathons. (Round your answer to four decimal places.) Part (c) Find the 80th percentile for the average of these 49 marathons. (Round your answer to two decimal places.) min Part (d) Find the median of the average running times. min

Q. A study was conducted of the Gouldian finch which lives in North Australia. Each bird observed was classified according to "Face Color" and "Bill colour" and the results are in the following table.

1) Reproduce the table with marginal and grand totals.

2) Estimate the proportion of the Gouldian finch population with:
-> Yellow face and a yellow bill
-> Yellow face OR a yellow bill
-> the same face and bill

3) Determine the probability that an individual Finch has:
a) A yellow face
b) a yellow face, given that it has a red bill

4) A member of the study observerd: "Clearly face color and Bill color are NOT independent ". Say if you agree or disagree with this with justification by using probabilities from above.

The medical community unanimously agrees on the health benefits of regular exercise, but are adults listening? During each of the past 15 years, a polling organization has surveyed americans about their exercise habits. In the most recent of these polls, slightly over half of all American adults reported that they exercise for 30 or more minutes at least three times per week. The following data show the percentages of adults who reported that they exercise for 30 or more minutes at least three times per week during each of the 15 years of this study.

Year/Percentage of Adults Who Exercise 30 or more minutes at least three times per week

1 41.9

2 45.5

3 47.3

4 45.8

5 46.7

6 44.7

7 47.9

8 50.4

9 48.4

10 49.3

11 49.6

12 52.6

13 50.6

14 55.1

15 52.4

a. Does a linear trend appear to be present?

b.use simple linear regression to find the parameters for the line that minimizes MSE for this time series. Do not round your interim computations and round your final answers to four decimal places. For subtractive or negative numbers use a minus sign. (Example: -300)

y-intercept, b0 =

Slope, b1 =

MSE =

c.Use the trend equation from part (b) to forecast the percentage of adults next year (year 16 of the study) who will report that they exercise for 30 or more minutes at least three times per week. Do not round your interim computations and round your final answers to four decimal places. For subtractive or negative numbers use a minus sign. (Example: -300)

_%

d.Use the trend equation from part (b) to forecast the percentage of adults three years from now (year 18 of the study) who will report that they exercise for 30 or more minutes at least three times per week. Do not round your interim computations and round your final answers to four decimal places. For subtractive or negative numbers use a minus sign. (Example: -300) %

Average yearly inflation. If an item costs C at one time and D n years later, and Cxⁿ = D, then we call x the average annual inflation factor (for example, x = 1.04 refers to an inflation rate of 4%).

At a 6% average annual inflation factor, it will take 12 years for the house to double.  If 6% is replaced by r%, bankers use 72/r to estimate the number of years. This is called the rule of 72. Graph this estimate and the actual answer over the interval 1 <= r <= 12. Comment on the accuracy of the rule of 72.

In a cereal box filling factory a randomly selected sample of 16 boxes have an average weight of 470 g with a standard deviation of 15g. The weights are normally distributed. Compute the 90% confidence interval for the weight of a randomly selected box. [I think I can compute the confidence interval but I'm being thrown off by the "randomly-selected box" requirement].

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

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?


State the null and alternate hypotheses.

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.   

  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 not 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?

Yes

No     

What sampling distribution will you use?

chi-square

Student's t     

normal

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 = 3.5. 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 exponential population distribution.     

We assume a uniform population distribution.

We assume a normal 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 < 0.010

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 below this interval.     

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

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

Two plots at Rothamsted Experimental Station were studied for production of wheat straw. For a random sample of years, the annual wheat straw production (in pounds) from one plot was as follows.

Use a calculator to verify that, for this plot, the sample variance is s² ≈ 0.325.

Another random sample of years for a second plot gave the following annual wheat production (in pounds).

Use a calculator to verify that the sample variance for this plot is s² ≈ 0.658.

Test the claim that there is a difference (either way) in the population variance of wheat straw production for these two plots. Use a 5% level of signifcance.

(a) What is the level of significance?

State the null and alternate hypotheses.

H_o: σ₁² = σ₂²; H₁: σ₁² > σ₂²

H_o: σ₁² > σ₂²; H₁: σ₁² = σ₂²     

H_o: σ₂² = σ₁²; H₁: σ₂² > σ₁²

H_o: σ₁² = σ₂²; H₁: σ₁² ≠ σ₂²

(b) Find the value of the sample F statistic. (Use 2 decimal places.)


What are the degrees of freedom?

What assumptions are you making about the original distribution?

The populations follow independent normal distributions.

The populations follow dependent normal distributions. We have random samples from each population.     

The populations follow independent normal distributions. We have random samples from each population.

The populations follow independent chi-square distributions. We have random samples from each population.

(c) Find or estimate the P-value of the sample test statistic. (Use 4 decimal places.)

p-value > 0.2000

.100 < p-value < 0.200     

0.050 < p-value < 0.1000

.020 < p-value < 0.0500

.002 < p-value < 0.020

p-value < 0.002

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.     

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

(e) Interpret your conclusion in the context of the application.

Fail to reject the null hypothesis, there is sufficient evidence that the variance in annual wheat production differs between the two plots.

Reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production differs between the two plots.     

Reject the null hypothesis, there is sufficient evidence that the variance in annual wheat production differs between the two plots.

Fail to reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production differs between the two plots.

Question 4 [25]
The Zambezi car battery manufacturer claims that the average lifespan of batteries produced by his firm is at least 30 months. The rival manufacture the Rehoboth batteries disagree and took random sample of 100 Zambezi car batteries and recorded a mean of 31.7 months and a standard deviation of 8 months. (Show all your works)
Determine the following:
a) The null and alternative hypotheses

b) The test statistic value.

c) The critical statistics value at 99% confidence level

d) The rejection region using critical value approach

e) The p value at 99% confidence level

f) The rejection region using both the p value approach

g) Make conclusion about the population mean using both approaches

Victoria’s Secret online offers 2,500 items for sale, 60 of them are offered to VIP’s only. Fredricks of Hollywood offers 1,300 items, where 45 are available to VIP members only. I believe that the proportion of VIP only items on the Fredricks website is more than the proportion of VIP items on the VS website.

Gather appropriate data and post the solution to compare these two proportions.

Find the critical value from the Studentized range distribution for H0: μ1 = μ2 = μ3 = μ4 = μ5, with n = 35 at α = 0.01. Round to the nearest 3 decimal places.

Question 3 [25]
OK furniture store submit weekly records the number of customer contacts contacted per week. A sample of 50 weekly reports showed a sample mean of 25 customer contacts per week. The sample standard deviation was 5.2. (Show all your works)
a) Compute the Margin of error at 0.05 significant level

b) Provide a 95% confidence interval for the population mean.

c) Compute the Margin of error at 0.01 significant level

d) Provide a 99% confidence interval for the population mean.

e) With a 0.99 probability, what size of sample should be taken if the desired margin of error is 1.5