Suppose we wish to generate a sample from the exponential ($\beta$) distribution, and only have access to a computer which generates numbers from the skew logistic distribution. It turns out that if $X$~SkewLogistic ($\beta$), then log(1+exp($-X$)) is exponential ($\beta$). Show that this is true and check by simulation that this transformation is correct.

The table below gives the age and bone density for five randomly selected women. Using this data, consider the equation of the regression line, yˆ=b0+b1xy^=b0+b1x, for predicting a woman's bone density based on her age. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant.

Step 4 of 6:

Substitute the values you found in steps 1 and 2 into the equation for the regression line to find the estimated linear model. According to this model, if the value of the independent variable is increased by one unit, then find the change in the dependent variable yˆy^.

Step 6 of 6:

Find the value of the coefficient of determination. Round your answer to three decimal places.

1. On average, do hospitals in the United States employ fewer than 900 personnel? Use the hospital database as your sample and an alpha of 0.10 to test this figure as the alternative hypothesis. Assume that the number of births and number of employees in the hospitals are normally distributed in the population.

2. Personnel
   792
   1762
   2310
   328
   181
   1077
   742
   131
   1594
   233
   241
   203
   325
   676
   347
   79
   505
   1543
   755
   959
   325
   954
   1091
   671
   300
   753
   607
   929
   354
   408
   1251
   386
   144
   2047
   1343
   1723
   96
   529
   3694
   1042
   1071
   1525
   1983
   670
   1653
   167
   793
   841
   316
   93
   373
   263
   943
   605
   596
   1165
   568
   507
   479
   136
   1456
   3486
   885
   243
   1001
   3301
   337
   1193
   1161
   322
   185
   205
   1224
   1704
   815
   712
   156
   1769
   875
   790
   308
   70
   494
   111
   1618
   244
   525
   472
   94
   297
   847
   234
   401
   3928
   198
   1231
   545
   663
   820
   2581
   1298
   126
   2534
   251
   85
   432
   864
   66
   556
   347
   239
   973
   439
   1849
   102
   262
   885
   549
   611
   330
   1471
   75
   262
   328
   377
   575
   1916
   2620
   571
   703
   535
   160
   202
   1330
   370
   3123
   2745
   815
   576
   502
   808
   50
   728
   4087
   3012
   68
   3090
   1358
   576
   284
   145
   2312
   1124
   336
   415
   1779
   338
   453
   437
   261
   609
   647
   61
   2074
   2232
   948
   409
   153
   741
   1625
   538
   789
   395
   956
   362
   144
   229
   396
   2256
   731
   1477
   102
   106
   939
   392
   3516
   785
   607
   273
   630
   1379
   1108
   583
   514
   216
   1593
   1055
   399
   834
   104

Sheila's doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational diabetes if the glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested. Sheila's measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with μ = 124 mg/dl and σ = 10 mg/dl. What is the level L such that there is probability only 0.05 that the mean glucose level of 2 test results falls above L for Sheila's glucose level distribution? (Round your answer to one decimal place.)

The following data represent the weight​ (in grams) of a random sample of 13 medicine tablets. Find the​ five-number summary, and construct a boxplot for the data. Comment on the shape of the distribution.

0.600 0.598 0.598 0.600 0.600 0.599 0.604 0.611 0.606 0.599 0.601 0.602 0.604  

find the five number summary

The weight of an organ in adult males has a​ bell-shaped distribution with a mean of 310 grams and a standard deviation of 20 grams. Use the empirical rule to determine the following.

​(a) About 95​% of organs will be between what​ weights?

​(b) What percentage of organs weighs between 250 grams and 370 grams?

​(c) What percentage of organs weighs less than 250 grams or more than 370 ​grams?

​(d) What percentage of organs weighs between 250 grams and 330 grams?

Price change to maximize profit. A business sells n products, and is considering changing the price of one of the products to increase its total profits. A business analyst develops a regression model that (reasonably accurately) predicts the total profit when the product prices are changed, given by Pˆ = βT x + P , where the n-vector x denotes the fractional change in the product prices, xi = (pnew − pi)/pi. Here P is the profit with the currentiprices, Pˆ is the predicted profit with the changed prices, pi is the current (positive) price of product i, and pnew is the new price of product i.

1. (a) What does it mean if β3 < 0? (And yes, this can occur.)

2. (b) Suppose that you are given permission to change the price of one product, by up to 1%, to increase total profit. Which product would you choose, and would you increase or decrease the price? By how much?

3. (c) Repeat part (b) assuming you are allowed to change the price of two products, each by up to 1%.

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

(a) Suppose n = 34 and p = 0.33.

(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/No    

second blank

can/cannot    

third blank

both n·p and n·q exceed

n·q exceeds

n·p exceeds

n·q does not exceed

n·p and n·q do not exceed

n·p 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/No    

second blank

can/cannot    

third blank

both n·p and n·q exceed

n·q exceeds

n·p exceeds

n·q does not exceed

n·p and n·q do not exceed

n·p does not exceed

fourth blank (Enter an exact number.)

(c) Suppose n = 52 andcp = 0.22.

(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/No    

second blank

can/cannot    

third blank

both n·p and n·q exceed

n·q exceeds

n·p exceeds

n·q does not exceed

n·p and n·q do not exceed

n·p 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 =

A die is tossed once, and the face, n, is noted. Then an integer m is selected at random from the set {1,2,···,n}, which depends on the face n. a). Find the probability that m = 3. b). Given that m = 3, what is the probability that n = 6?

1. How many 12-digit phone numbers can be created with the following restrictions:

a) no restrictions

b) first number cannot be zero or one.

c) no repeated numbers

2. Find the number of distinguishable permutations of the letters in the following words.

a) CALCULUS

b) PEPPER

c) MISSISSIPPI

Based on the model ​N(1153​,85​) describing steer​ weights, what are the cutoff values for

​a) the highest​ 10% of the​ weights?

​b) the lowest​ 20% of the​ weights?

​c) the middle​ 40% of the​ weights?

Write down a brief report of the results from this regression analysis explaining: (1) what is the impact of each variable over the demand? (2) How strong are the results from this analysis to support a forecast? (3) What are the limitations you foresee by using this analysis to forecast production for the following five years?

Introduction to Probability and Statistics

Scenario: We wish to compare the commuting time in minutes to the university of two sections of a particular

Morning Section Times:
39 35 39 39 40 37 41 39 42 40 37 35 38 36 40 35 38 36 39 35 38 35 39 38 41 39 38 40 38 41 41 37 34 41 37 41 35 39 36 41

Evening Section Times:
35 47 29 34 26 34 38 45 44 49 37 37 37 37 40 26 29 30 23 38 32 36 45 39 31 42 41 35 34 43 31 30 37 36 33

Part 1 Create one side-by-side boxplot of the two sets of times (i.e. both boxplots on the same axes). The axes for the boxplots should have appropriate labels. Copy and paste this boxplot into your document. The boxplots themselves may be either horizontal or vertical.

Part 2 Use R to calculate the sample mean and sample standard deviation of the times for the two sections. Copy and paste the relevant commands and output from the R Console Window into your document.

Part 3 In your opinion, which class appears to have the longer commute times? Write a few sentences explaining your opinion. You should make reference to the relevant features of the two data sets (e.g. the sample mean or median, the spread of the data, minimum/maximum values, etc.)

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

What's your favorite ice cream flavor? For people who buy ice cream, the all-time favorite is still vanilla. About 20% of ice cream sales are vanilla. Chocolate accounts for only 9% of ice cream sales. Suppose that 185 customers go to a grocery store in Cheyenne, Wyoming, today to buy ice cream. (Round your answers to four decimal places.)

(a) What is the probability that 50 or more will buy vanilla?

(b) What is the probability that 12 or more will buy chocolate?

(c) A customer who buys ice cream is not limited to one container or one flavor. What is the probability that someone who is buying ice cream will buy chocolate or vanilla? Hint: Chocolate flavor and vanilla flavor are not mutually exclusive events. Assume that the choice to buy one flavor is independent of the choice to buy another flavor. Then use the multiplication rule for independent events, together with the addition rule for events that are not mutually exclusive, to compute the requested probability.

(d) What is the probability that between 50 and 60 customers will buy chocolate or vanilla ice cream? Hint: Use the probability of success computed in part (c).