A person's blood glucose level and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean μ = 81 and standard deviation σ = 26. Note: After 50 years of age, both the mean and standard deviation tend to increase. For an adult (under 50) after a 12-hour fast, find the following probabilities. (Round your answers to four decimal places.)

(a) x is more than 60

(b) x is less than 110

(c) x is between 60 and 110

(d) x is greater than 125 (borderline diabetes starts at 125)

7.26. An appliance manufacturer offers maintenance contracts on its major appliances. A manager wants to know what fraction of buyers of the company’s convection ovens are also buying the maintenance contract with the oven. From a random sample of 120 sales slips, 31 of the oven buyers opted for the contract.

a. The proportion of customers who buy the contract along with their oven is estimated as ____?

b. Calculate a standard error for the estimate in part a.

c. Calculate a 95% interval estimate for the true proportion of customers who buy the contract along with their oven.

d. Interpret the interval in part c.

The table below lists the number of games played in a yearly​ best-of-seven baseball championship​ series, along with the expected proportions for the number of games played with teams of equal abilities. Use a 0.05 significance level to test the claim that the actual numbers of games fit the distribution indicated by the expected proportions.

Determine the null and alternative hypotheses.

Upper H 0H0​:

▼

Upper H 1H1​:

▼

Calculate the test​ statistic,

chi squaredχ2.

chi squaredχ2equals=nothing

​(Round to three decimal places as​ needed.)

Calculate the​ P-value.

​P-valueequals=nothing

​(Round to four decimal places as​ needed.)

What is the conclusion for this hypothesis​ test?

A.

RejectReject

Upper H 0H0.

There is

sufficientsufficient

evidence to warrant rejection of the claim that the actual numbers of games fit the distribution indicated by the expected proportions.

B.

Fail to rejectFail to reject

Upper H 0H0.

There is

insufficientinsufficient

evidence to warrant rejection of the claim that the actual numbers of games fit the distribution indicated by the expected proportions.

C.

Fail to rejectFail to reject

Upper H 0H0.

There is

sufficientsufficient

evidence to warrant rejection of the claim that the actual numbers of games fit the distribution indicated by the expected proportions..

D.

RejectReject

Upper H 0H0.

There is

insufficientinsufficient

evidence to warrant rejection of the claim that the actual numbers of games fit the distribution indicated by the expected proportions.

Imagine you are selected as a contestant on The Price is Right. The host, Bob Barker, shows you three boxes with marbles in them. Box #1 contains 6 red, 2 white and 2 blue marbles. Box #2 contains 2 red, 5 white and 3 blue marbles. Lastly, Box #3 contains 5 red, 2 white and 3 blue marbles. One of Bob's assistants will pick marbles from one of the boxes after they are hidden behind a curtain. If you correctly guess which box the marbles were picked from, you win a brand new car!

c. Suppose one red, one white, and one blue marble are selected, without replacement. Calculate the likelihood of the sample if they are selected from Box 1, 2 and 3.

P(sample | Box 1) =  ,     P(sample | Box 2) =  ,     P(sample | Box 3) =  .

Which box should you pick to maximize your chance of winning a car? (select more than one if there are ties)
Box 1
Box 2
Box 3

1. Suppose you are studying the effects of a new drug on diastolic blood pressure. Twelve patients were enrolled in the study. The patients were asked to takethe old drug for 6 months, and then the new drug for the remaining 6 months. The blood pressure measurements are given in the following table.

1. (0.5 p)Are the data paired or unpaired? Justify your answer.
2. (0.5 p)Find the 95% confidence interval for the difference of the means.
3. (1 p)Perform the hypothesis test (significance level α = 0.05). Is the new drug more effective compared to the old drug?

Part (1)

Consider the data in the above table for number of pets in 100 households. Do the following:

1

• – Use the R command chisq.test to test whether the data can be fitted by a Poisson model with α = 0.05. Report the p-value and the number of degrees of freedom from your test.

• – Are there any corrections necessary and if so why?

• – How would you update the code to include these corrections and what is

  the new p-value obtained?

  Use the R-builtin function dpois to calculate the Poisson probabilities.

• Part (2)

  Run install.packages(”MASS”) to install the MASS library. Do the follow- ing:

  • – For the data frame survey in this package first remove any NA values with the command survey = na.omit(survey). Following that consider the 2 categorical variables Sex and Exer in survey. How many different types of values does Exer take?

  • – Use the R command chisq.test to test whether there is an association between these variables with α = 0.05. Report the p-value obtained.

  • – Remove the level Freq for the variable Exer and report the p-value ob- tained by running chisq.test between Sex and Exer.

1a) Explain why we fail to reject the null hypothesis when the p-value is greater than the level of significance.

b) If the null hypothesis is rejected at a level of significance of 5%, does it automatically get rejected at a level of significance of 1%? Explain your reasoning with an example using a p-value(s) that you make up for different scenarios.

c) If the null hypothesis is rejected at a level of significance of 1%, does it automatically get rejected at a level of significance of 5%? Explain your reasoning with an example using a p-value(s) that you make up for different scenarios.

1. As a physical education teacher, you want to try a new way of teaching students to run in order to improve physical fitness scores. You plan to try the new plan with one class and compare their mean score with fitness scores across the state.
   1. What are your alternative and null hypotheses for a non-directional test?
   2. What are your alternative and null hypotheses for a directional test?
   3. Would you use a non-directional or a directional test? Why?
   4. For your choice of test, what are the two possible outcomes of correct decisions?
   5. What are the outcomes of Type I and Type II errors?
   6. Draw a decision table with the four possible outcomes written on it.
   7. Select a level of alpha and discuss why you chose that level.
   8. Draw a curve that shows your null hypothesis for the test you chose and alpha level.

Suppose you are playing a game with a friend in which you bet ? dollars on the flip of a fair coin: if the coin lands tails you lose your ? dollar bet, but if it lands heads, you get 2? dollars back (i.e., you get your ? dollars back plus you win ? dollars).

Let ? = "the amount you gain or lose."

(a) What is the expected return ?(?) on this game? (Give your answer in terms of ?)

Now, after losing a bunch of times, suppose you decide to improve your chances with the following strategy: you will start by betting $1, and if you lose, you will double your bet the next time, and you will keep playing until you win (the coin has to land heads sometime!).

Let ? = "the amount you gain or lose with this strategy".

(b) What is the expected return ?(?) with this strategy? (Hint: think about what happens for each of the cases of ?=1,2,3… flips).

(c) Hm ... do you see any problem with this strategy? How much money would you have to start with to guarantee that you always win?

(d) Suppose when you apply this strategy, you start with $20 and you quit the game when you run out of money. Now what is ?(?)?

A. Calculate the Tidal Volumes

B. What are the standard deviations for males and females?

Females Data

Males Data

1.One of the relatively uncommon instances in which researchers know the population standard deviation is in the case of the Intelligence Quotient or IQ test. In general, the average IQ score in large, diverse populations is 100 and the standard deviation is 15. Suppose that your sample of 300 members of your community gives you a mean IQ score of 108. Calculate a 90% confidence interval for the mean and indicate which answers come closest to those that would fill the blanks in the following interpretation: we can be 90% confident that they mean IQ score in this community lies between _____ and _____ .

2.Suppose that you are a city planner who obtains and sample of 20 randomly selected members of a mid-sized town in order to determine the average amount of money that residents spend on transportation each month (such as fuel, vehicle repairs, and public transit). You do not have the population standard deviation. To 3 decimal places, what is the critical value for the 95% confidence interval? In the same scenario as 7.09, suppose you obtained a mean of $167 spent on transportation and a standard deviation of $40. Calculate a 95% confidence interval for the mean and select the values that come closest to those that would fill the spaces in the following interpretation: we can be 95% confident that they mean amount of money spent on transportation lies between _____ and _____ .

(Combinations -- Poker Probabilities) Suppose you deal a poker hand of 5 cards from a standard deck

(a) What is the probability of a flush (all the same suit) of all red cards?

(b) What is the probability of a full house where the 3-of-a-kind include two black cards, and the 2-of-a-kind are not clubs?

(c) What is the probability of a pair, where among the 3 non-paired cards, we have 3 distinct suits?

Hint: These are straight-forward modifications of the formulae given in lecture. You may quote formulae already given in lecture or lab without attribution and without explaining how to get the formula.

11.Private Four-Year College Enrollment A random sample of enrollments in Pennsylvania’s private four-year colleges is listed here. Check for normality. Answer: Not Normal. Please show work this is a review for an exam

Here is a bivariate data set.

Find the correlation coefficient and report it accurate to four decimal places.
r =