Kilgore's Deli is a small delicatessen located near a major university. Kilgore does a large walk-in carry-out lunch business. The deli offers two luncheon chili specials, Wimpy and Dial 911. At the beginning of the day, Kilgore needs to decide how much of each special to make (he always sells out of whatever he makes). The profit on one serving of Wimpy is $0.35, on one serving of Dial 911, $0.59. Each serving of Wimpy requires 0.45 pound of beef, 0.45 cup of onions, and 5 ounces of Kilgore's special sauce. Each serving of Dial 911 requires 0.45 pound of beef, 0.72 cup of onions, 1 ounces of Kilgore's special sauce, and 4 ounces of hot sauce. Today, Kilgore has 33 pounds of beef, 47 cups of onions, 81 ounces of Kilgore's special sauce, and 48 ounces of hot sauce on hand.

1. Develop an LP model that will tell Kilgore how many servings of Wimpy and Dial 911 to make in order to maximize his profit today.

Let

W = # of servings of Wimpy to make

D = # of servings of Dial 911 to make

b. Find an optimal solution. Truncate your answers to whole servings available for sale.

Solution:

W = 13

D = 12

1. What is the optimal profit? Round your answer to the nearest cent.

   Profit = $ __________

2. What is the dual value for special sauce? Round your answer to the nearest cent.
   
   Dual value for special sauce = $ 0.07
   
3. Increase the amount of special sauce available by 1 ounce. Give the new solution. Round the answer for profit to the nearest cent.
   
   Solution: W = ________ , D = 12 , Profit = $ _________
   
   Does the solution confirm the answer to part (c)?
   
   Dual value is confirmed.

1. There is variation in the migratory behavior of certain bird species. Some populations migrate while others do not. A study was run on Blue-Tailed Bee-Eaters. The south-east Asia population migrates south for the winter while the Australasia population does not. Using a capture technique called mist-netting, wing lengths were measured for each population. Is there a statistically significant difference between the two populations? What conclusion can you draw from this information?

Use your reading handout to calculate all the necessary parts of a t-test. Use an alpha=0.25 in the t-table to find your critical value.

Population A                                                                                             Population B

(South-East Asia, Migratory)                                                       (Australasia, Non-migratory)

Wing length (cm)                                                                         Wing length (cm)

      31                                                                                                        47

      50                                                                                                        41

      30                                                                                                        41

      31                                                                                                        37

      42                                                                                                        30

      37                                                                                                        34

      50                                                                                                        35

      45                                                                                                        31

      41                                                                                                        33

      50                                                                                                        31

      45                                                                                                        37

      31                                                                                                        47

      48                                                                                                        44

      39                                                                                                        33                                      47                                                                                                  35

      48                                                                                                        37

      50

47

46

39

42

46

1.Historical records show that the average per capita consumption of eggs in the US is 278 (this is per year). Lately, consumers have been getting conflicting advice about the benefits and dangers of eggs, and a rural agricultural association wants to know if the average egg consumption has changed. A sample of 50 adults yielded a sample mean egg consumption of 287 eggs and a sample standard deviation of 28 eggs. Construct a null and alternate hypothesis that the association can use to answer their question. Based on the sample data and a 5% level of significance, what can they conclude?

2. In 2010, the average weight of cell phone batteries was 110 grams. Battery technology has continued to improve and a consumer group wants to know if the average weight of cell phone batteries has reduced. A sample of 49 cell phone batteries yielded a sample mean of 108 grams. Historical data shows that the population standard deviation is 10 grams. At the 5% level of significance, can the consumer group conclude that batteries have become lighter? Would your answer change at a 1% level of significance? Justify.

Use the data to assess the correlation between hours studying per week in High School and SAT score. Interpret the result and give the conclusion.

In an effort to promote a new product, a marketing firm asks participants to rate the effectiveness of ads that varied by length (short, long) and by type of technology (static, dynamic, interactive). Higher ratings indicated greater effectiveness.

(a) Complete the F-table and make a decision to retain or reject the null hypothesis for each hypothesis test. (Assume experimentwise alpha equal to 0.05.)

State the decision for the main effect of length.

Retain the null hypothesis.Reject the null hypothesis.     

State the decision for the main effect of technology.

Retain the null hypothesis.Reject the null hypothesis.     

State the decision for the interaction effect.

Retain the null hypothesis.Reject the null hypothesis.     

(b) Based on the results you obtained, what is the next step?

No further analysis is needed, because none of the effects are significant.Compute pairwise comparisons for the technology factor.      Compute simple main effect tests for the significant interaction.Compute pairwise comparisons for the length factor.

The mean height of adult males in the U.S is about 68.8" with a variance of 11.56". Suppose the mean height of a sample of 31 mentally disabled males was found to be 67.3". Researchers want to know if the height of mentally disabled males differs from the population. What can be concluded with α = 0.01?

Compute the appropriate test statistic(s) to make a decision about H₀.
critical value =  

If appropriate, compute the CI. If not appropriate, input "na" for both spaces below.
[   ,   ]

e) Compute the corresponding effect size(s) and indicate magnitude(s).
d =   ;   ---Select--- na trivial effect small effect medium effect large effect

On a planet far far away from Earth, IQ of the ruling species is normally distributed with a mean of 105 and a standard deviation of 14. Suppose one individual is randomly chosen. Let X = IQ of an individual.

a. What is the distribution of X? X ~ N(,)

b. Find the probability that a randomly selected person's IQ is over 97. Round your answer to 4 decimal places.

c. A school offers special services for all children in the bottom 4% for IQ scores. What is the highest IQ score a child can have and still receive special services? Round your answer to 2 decimal places.

d. Find the Inter Quartile Range (IQR) for IQ scores. Round your answers to 2 decimal places.

Q1:

Q3:

IQR:

Suppose we have a binomial experiment in which success is defined to be a particular quality or attribute that interests us.

(a)

• Suppose n = 25 and p = 0.28.

(For each answer, enter a number. Use 2 decimal places.)
n·p = ?
n·q = ?

Can we approximate p̂ by a normal distribution? Why? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank

Yes or No    

second blank

can or cannot    

third blank

n·p exceedsn·q exceeds
n·p does not exceed
n·p and n·q do not exceed
both n·p and n·q exceed
n·q does not exceed

fourth blank (Enter an exact number.)
?

What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ = mu sub p hat = ?

σ_p̂ = sigma sub p hat = ?

(b)

Suppose

• n = 25 and p = 0.15.

Can we safely approximate p̂ by a normal distribution? Why or why not? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank

Yes or No    

second blank

can or cannot    

third blank

n·p exceedsn·q exceeds
n·p does not exceed
n·p and n·q do not exceed
both n·p and n·q exceed
n·q does not exceed

fourth blank (Enter an exact number.)
?

(c)

Suppose

• n = 48 and p = 0.40.

(For each answer, enter a number. Use 2 decimal places.)
n·p = ?
n·q = ?

Can we approximate p̂ by a normal distribution? Why? (Fill in the blank. There are four answer blanks. A blank is represented by _____.)

_____, p̂ _____ be approximated by a normal random variable because _____ _____.

first blank

Yesor No    

second blank

can or cannot    

third blank

n·p exceedsn·q exceeds
n·p does not exceed
n·p and n·q do not exceed
both n·p and n·q exceed
n·q does not exceed

fourth blank (Enter an exact number.)
?

What are the values of μ_p̂ and σ_p̂? (For each answer, enter a number. Use 3 decimal places.)
μ_p̂ = mu sub p hat =

σ_p̂ = sigma sub p hat =

For the data set

(a) Find the 76th percentile.

(b) Find the 42nd percentile.

(c) Find the 16th percentile.

(d) Find the 65th percentile.

Suppose x has a normal distribution with a mean of 79 and a variance of 441.00. If a sample of 15 were randomly drawn from the population, find the probability of   mu hat   for each of the following situations.

a) less than 77: probability =

b) greater than 83: probability =

c) in between 65 and 76: probability =

d) in between 76 and 94: probability =

Suppose 31 pregnant women are sampled who smoke an average of 22 cigarettes per day with a variance of 144.00.

a) What is the probability that the pregnant women will smoke an average of 20 cigarettes or more? probability =

b) What is the probability that the pregnant women will smoke an average of 21 cigarettes or less? probability =

c) What is the probability that the pregnant women will smoke an average of 18 to 24 cigarettes? probability =

d) What is the probability that the pregnant women will smoke an average of 23 to 26 cigarettes? probability =

Note: Do NOT input probability responses as percentages; e.g., do NOT input 0.9194 as 91.94.

A researcher wishes to test the effects of excerise on the ability to complete a basic skills test. He designs a pre-test and post-test to give to each participant. You believe that there was an increase in the scores. You believe the population of the differences is normally distributed, but you do not know the standard deviation. When calculating difference use Post-test minus Pre-test. pre-test post-test 52 60 60 55 44 49 92 94 84 76 55 65 64 58 67 66 53 59 99 99 75 79 77 82 Which of the following are the correct hypotheses? H 0 : μ d ≥ 0 H 0 : μ d ≥ 0 H A : μ d < 0 H A : μ d < 0 (claim) H 0 : μ d ≤ 0 H 0 : μ d ≤ 0 H A : μ d > 0 H A : μ d > 0 (claim) H 0 : μ d = 0 H 0 : μ d = 0 H A : μ d ≠ 0 H A : μ d ≠ 0 (claim) Correct

Given that α α is 0.10 the critical value is 1.363

The test statistic is: Incorrect(round to 3 places)

The p-value is: Incorrect(round to 3 places)

In an effort to improve the mathematical skills of 18 students, a teacher provides a weekly 1-hour tutoring session. A pre-test is given before the sessions and a post-test is given after. The results are shown here. Test the claim that there was an increase in the scores. at αα=0.01. You believe that the population is normally distributed, but you do not know the standard deviation. When calculating difference use Post-test minus Pre-test.

Which of the following are the correct hypotheses?

• H0:μd≥0H0:μd≥0
  HA:μd<0HA:μd<0(claim)
• H0:μd≤0H0:μd≤0
  HA:μd>0HA:μd>0(claim)
• H0:μd=0H0:μd=0
  HA:μd≠0HA:μd≠0(claim)

Given that αα is 0.01 the critical value is 2.567
The test statistic is: (round to 3 places)
The p-value is: (round to 3 places)

Suppose that the national average for the math portion of the College Board's SAT is 518. The College Board periodically rescales the test scores such that the standard deviation is approximately 50. Answer the following questions using a bell-shaped distribution and the empirical rule for the math test scores. If required, round your answers to two decimal places. (a) What percentage of students have an SAT math score greater than 568? % (b) What percentage of students have an SAT math score greater than 618? % (c) What percentage of students have an SAT math score between 468 and 518? % (d) What is the z-score for student with an SAT math score of 620? (e) What is the z-score for a student with an SAT math score of 405?

Your claim results in the following alternative hypothesis:
H_a : μ ≠≠ 166
which you test at a significance level of α=.02α=.02.

Find the positive critical value, to three decimal places.

z_α/2 =

You are performing a right-tailed t-test with a sample size of 5

If α=.005α=.005, find the critical value, to two decimal places.

You are performing a two-tailed test.

If α=.06α=.06, find the positive critical value, to three decimal places.

z_α/2 =

Testing:

H0:μ=7.83H0:μ=7.83
H1:μ>7.83H1:μ>7.83

Your sample consists of 23 subjects, with a mean of 8.3 and a sample standard deviation (s) of 4.14.

Calculate the test statistic, rounded to 2 decimal places.

t=t=

You are performing a two-tailed z-test

If α=0.1α=0.1, find the positive critical value, to two decimal places.

Using R Studio

Now, set the seed to 348 with `set.seed()`. Then take a sample of size 10,000 from a normal distribution with a mean of 82 and a standard deviation of 11.

(a) Using sum() on a logical vector, how many draws are less than 60? Using mean() on a logical vector, what proportion of the total draws is that? How far is your answer from pnorm() in 1.1 above?

```{R}
set.seed(348)
x=rnorm(10000,82,11)
sum(ifelse(x<60,1,0))

mean(ifelse(x<60,1,0))

pnorm(60,82,11)

Using sum() function there are 128 draws that are less than 60 and using the mean() function 0.0281 is the porportion of total draws. From these outputs we can say that the answer is quite close to the pnorm() value that has been calculated.

(b) What proportion of your sample is greater than 110 or less than 54?

(c) Why are your answers close to what you got above? Why are they not exactly the same?

(d) Using ggplot2, make a histogram of your sample. Set y=..density.. inside aes(). Overlay a normal distribution with stat_function(aes(samp), fun=dnorm, args=list(82,11)). Using geom_vline(xintercept=), add dashed vertical lines corresponding to the 2.5th and the 97.5th percentile of the sample