2. Suppose that the probability that a grant proposal is awarded by a funding agency is 0.3. (a) If, for a particular year, there are 100 proposals submitted to that agency, what is the probability that at most 20 proposals are awarded? Consider any necessary approximation.

The distribution of weights for 12 month old baby boys in the US is approximately normal with mean 22.5 pounds and a standard deviation of 2.2 pounds. a. if a 12 month old boy weights 20.3 pounds, what weight percentile is he in approximately. b. if a 12 month old boy is in the 84th percentile in weight, estimate his weight. c. Estimate the weight of a 12 month old boy who is in the 25th percentile by weight. d. Estimate the weight of a 12 month old boy who is in the 75 percentile by weight.

A quality characteristic X follows a Normal distribution with mean 100 and standard deviation 2 when the production process is in control.  

Part (a): Suppose that the lower and upper specification limits are 94 and 106, respectively.  What is the percentage of produced product items that are non-conforming, i.e., do not conform to the specification limits, when the process is indeed in control?

Part (b): The management has decided to monitor the process with and R control charts with a sample size of 4.  What should be the lower limit, centerline and the upper limit for each of the two control charts?

1. (18.10) Emissions of sulfur dioxide by industry set off chemical changes in the atmosphere that result in "acid rain." The acidity of liquids is measured by pH on a scale of 0 to 14. Distilled water has pH 7.0, and lower pH values indicate acidity. Normal rain is somewhat acidic, so acid rain is sometimes defined as rainfall with a pH below 5.0. Suppose that pH measurements of rainfall on different days in a Canadian forest follow a Normal distribution with standard deviationσ= 0.5. A sample ofndays finds that the mean pH isx= 4.8.

Give a 80 % confidence interval for the mean pH μ when n = 5, n = 15, and n = 40

n= 5 ____to ____

n= 15__ to ___

n= 40 __ to __

Recall that Benford's Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "1" as the leading digit is about 0.301. Now suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say you took a random sample of n = 216 numerical entries from the file and r = 52 of the entries had a first nonzero digit of 1. Let p represent the population proportion of all numbers in the corporate file that have a first nonzero digit of 1.

(i) Test the claim that p is less than 0.301. Use α = 0.10.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p = 0.301; H₁: p < 0.301

H₀: p = 0.301; H₁: p ≠ 0.301    

H₀: p < 0.301; H₁: p = 0.301

H₀: p = 0.301; H₁: p > 0.301

(b) What sampling distribution will you use?

The Student's t, since np < 5 and nq < 5.

The standard normal, since np > 5 and nq > 5.    

The Student's t, since np > 5 and nq > 5.

The standard normal, since np < 5 and nq < 5.

What is the value of the sample test statistic? (Round your answer to two decimal places.)


(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)


Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.10 level, we reject the null hypothesis and conclude the data are not statistically significant.    

At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

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

There is sufficient evidence at the 0.10 level to conclude that the true proportion of numbers with a leading 1 in the revenue file is less than 0.301.

There is insufficient evidence at the 0.10 level to conclude that the true proportion of numbers with a leading 1 in the revenue file is less than 0.301.    

(ii) If p is in fact less than 0.301, would it make you suspect that there are not enough numbers in the data file with leading 1's? Could this indicate that the books have been "cooked" by "pumping up" or inflating the numbers? Comment from the viewpoint of a stockholder. Comment from the perspective of the Federal Bureau of Investigation as it looks for money laundering in the form of false profits.

No. The revenue data file seems to include more numbers with higher first nonzero digits than Benford's law predicts.

No. The revenue data file does not seem to include more numbers with higher first nonzero digits than Benford's law predicts.   

Yes. The revenue data file seems to include more numbers with higher first nonzero digits than Benford's law predicts.

Yes. The revenue data file does not seem to include more numbers with higher first nonzero digits than Benford's law predicts.

(iii) Comment on the following statement: If we reject the null hypothesis at level of significance α, we have not proved H_o to be false. We can say that the probability is α that we made a mistake in rejecting H_o. Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud?

We have not proved H₀ to be false. Because our data lead us to reject the null hypothesis, more investigation is merited.

We have proved H₀ to be false. Because our data lead us to reject the null hypothesis, more investigation is not merited.    

We have not proved H₀ to be false. Because our data lead us to reject the null hypothesis, more investigation is not merited.

We have not proved H₀ to be false. Because our data lead us to accept the null hypothesis, more investigation is not merited.

The following relative frequency distribution was constructed from a population of 550. Calculate the population mean, the population variance, and the population standard deviation. (Negative values should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.)

Question 12 Unsaved A sample of University of Colorado students each viewed one of two simulated news reports about a terrorist bombing against the United States by a fictitious country. One report showed the bombing attack on a military target and the other on a cultural/educational site. Additionally, before viewing the news report, each student read one of two "primes." The first was a prime for forgiveness based on the biblical saying "Love thy enemy," while the second was a retaliatory prime based on the biblical saying "An eye for an eye, and a tooth for a tooth." After viewing the news report, the students were asked to rate on a scale of 1 to 12 what the U.S. reaction should be, with the lowest score (1) corresponding to the United States sending a special ambassador to the country and the highest score (12) corresponding to an all-out nuclear attack against the country.6 (Use a diagram like Figure 9.2 from the text to display the factors and treatments.) Identify the following in this experiment:

_____ eye-for-an-eye prime

_____ the students

_____ love thy neighbor prime

_____ rating of U.S. reaction to attack

_____ prime used

_____ cultural/educational target

_____ military target

_____ type of attack

Options:

1. Subjects

2. Factors

3. Treatments for the prime

4. Treatments for the type of attack

5. Response variable

Exercise 8-31 Algo The monthly closing stock prices (rounded to the nearest dollar) for Panera Bread Co. for the first six months of 2010 are reported in the following table. [You may find it useful to reference the t table.] Months Closing Stock Price January 145 February 144 March 149 April 146 May 150 June 140 Source: http://finance.yahoo.com.

a. Calculate the sample mean and the sample standard deviation. (Round intermediate calculations to at least 4 decimal places and "Sample mean" and "Sample standard deviation" to 2 decimal places.)

b. Calculate the 90% confidence interval for the mean stock price of Panera Bread Co., assuming that the stock price is normally distributed. (Round "t" value to 3 decimal places and final answers to 2 decimal places.)

Consider the following regression output with Sunday circulation of newspapers as dependent variable and Daily circulation as independent variable. Both Sunday and Daily circulation and measured in thousands of copies.

Given the output above choose whether the following statement is TRUE or FALSE.

Question 1
This regression is bad because we are only 39.8% confident that the intercept coefficient is not 0.

1. Quinnipiac University conducted a telephone survey with a randomly selected national sample of 1155 registered voters. The survey asked the respondents, “In general, how satisfied are you with the way things are going in the nation today?” (Quinnipiac University, February 7, 2017). Response categories were Very satisfied, Somewhat satisfied, Somewhat dissatisfied, Very dissatisfied, Unsure/No answer. a. What is the relevant population? b. What is the variable of interest? Is it qualitative or quantitative? c. What is the sample and sample size? d. What is the inference of interest to Gallup; that is, what are they trying to measure or learn about? e. What method of data collection is employed? f. How likely is the sample to be representative? g. Would it make more sense to use averages or percentages as a summary of the data for this question? h. Of the respondents, 24% said they felt somewhat satisfied. How many individuals provided this response?

find the sample size needed to give a margin of error to estimate a proportion within plus minus 2% within 99% confidence within 95% confidence within 90% confidence assume no prior knowledge about the population proportion p

Use R to complete the following questions. You should include your R code, output and plots in your answer.

1. Two methods of generating a standard normal random variable are:

a. Take the sum of 5 uniform (0,1) random numbers and scale to have mean 0 and standard deviation 1. (Use the properties of the uniform distribution to determine the required transformation).

b. Generate a standard uniform and then apply inverse cdf function to obtain a normal random variate (Hint: use qnorm).

For each method generate 10,000 random numbers and check the distribution using

a. Normal probability plot

b. Mean and standard deviation

c. The proportion of the data lying within the theoretical 2.5 and 97.5 percentiles and the 0.5 and 99.5 percentiles. (Hint: The ifelse function will be useful)  

These problems may be solved using Minitab. Copy and paste the appropriate Minitab output into a word-processed file. Add your explanations of the output near the Minitab output. DO NOT SIMPLY ATTACH PAGES OF OUTPUT AS AN APPENDIX.

Each problem should be able to fit on one or two pages, and each problem should include the following:

• Minitab output for the ANOVA.
• Written statement interpreting the ANOVA.
• Four-in-one plot of the residuals.
• Written interpretation as to whether the three assumptions of the ANOVA were met.
• Tukey comparisons if necessary.
• A written summary of your interpretation of the analysis in terms of the problem. This may involve more than one statement. In other words, state which group is “best” in terms of the problem.

1. Battery life for competing smartphone designs are being studied. 5 models of phone were fully charged and set to play the same video on repeat (the volume and brightness levels were to the same levels using laboratory instruments). The test was replicated 10 times for each phone, with the time until the phone turned off recorded (in hours). Analyze this problem as a CRD ANOVA to determine if the phone models differ in their battery life, and how they differ. If they are necessary, perform Tukey comparisons using Minitab AND BY HAND.

2. Lactation promotes a temporary loss of bone mass to provide adequate

amounts of calcium for milk production. Consider the data on total

body bone mineral content for a sample both during lactation (L) and

in the post weaning period (P).

1 . 2    3 4 5    6    7 8    9    10

L 1928 2549 2825 1924 1628 2175 2114 2621 1843 2541

P 2126 2885 2895 1942 1750 2184 2164 2626 2006 2627

Does the data suggest that true average total body bone mineral con-

tent during post weaning exceeds that during lactation by more than

25g? Use results from an output you obtained from the R software to

state and test the appropriate hypotheses with 95% confidence level(show

all details of R, either print our the R code and result or hand-write

R code and result ). State any assumptions required for the test to be

valid.

5. The following table summarizes the skin colours and position of 368

NBA players in 2014. Suppose that an NBA player is randomly selected

from that years player pool.

2

Guard positionForward Center Total

white 26 30    28 84

skin colour black 128 122    34    284

Total 154 152    62    368

(a) Find out where the variable "skin colour" is independent with the

variable "position" with

alpha = 0.05

(b) We only concern about the variable "position". The media claims

that the proportion of "Guard" is the same as the proportion of "For-

ward", which is twice as the proportion of "Center". Conduct a test to

find out whether the statement is valid with 90% confidence interval.

(hint: to test H0: P1= 0.4, P2 = 0.4, P3= 0.2)

6. Complete the following ANOVA table ( find the value of ?1, ?2, ?3, ?4,

?5) [5] and give the null hypothesis and the alternative hypothesis, [2]

give your conclusion based on the ANOVA table [2].

ANOVA table

Df SumSq Mean Sq F value P value

brands 3 39.757 ?1 ?5 5.399e-07

Error    36 ?2 ?3

Total    ?4 . 68.128

7. Refer to the

Bulletin of Marine Science (April 2010)

study of teams

of shermen shing for the red apiny lobster in Baja Valifornia Sur,

Mexico. Two variables measured for each of 8 teams from the Punta

Abreojos shing cooperative were y=total catch of lobsters (in kilo-

grams) during the season and x=average percentage of traps allocated

per day to exploring areas of unknown catch (called search frequency)

total catch search frequency

2785 35

6535 21

6695    26

4891    29

4937    23

5727    17

7019 21

5735 20

(a) Graph the data in a scatterplot (using R). What type of trend, if any,

could be observed?

(b) Add the regression line to the plot (using R, either hand-write the R

code and plot the graph or print your R code and graph).

(c)Give the null and alternative hypothesis for testing whether total catch

is negatively linearly related to search frequency. Find the p-vale of the test

and give the appropriate conclusion of the test using alpha = 0.05

(d) what's the coefficient of correlation between total catch and search

frequency?

What are the advantage and disadvantage of assuming quadratic utility functions in mean variance analysis?