# = 8

1. Suppose next that we have even less knowledge of our patient, and we are only given the accuracy of the blood test and prevalence of the disease in our population. We are told that the blood test is 9# percent reliable, this means that the test will yield an accurate positive result in 9#% of the cases where the disease is actually present. Gestational diabetes affects #+1 percent of the population in our patient’s age group, and that our test has a false positive rate of #+4 percent. Use your knowledge of Bayes’ Theorem and Conditional Probabilities to compute the following quantities based on the information given only in part 2:
   1. If 100,000 people take the blood test, how many people would you expect to test positive and actually have gestational diabetes?
   2. What is the probability of having the disease given that you test positive?
   3. If 100,000 people take the blood test, how many people would you expect to test negative despite actually having gestational diabetes?
   4. What is the probability of having the disease given that you tested negative?
   5. Comment on what you observe in the above computations. How does the prevalence of the disease affect whether the test can be trusted?

Fill in the conditional probability table here, then answer the questions in each part below.

1. Answer part (a) here.

2. Answer part (b) here.

3. Answer part (c) here.

4. Answer part (d) here.

5. Comment on how prevalence of the disease affects your ability to trust the test. Discuss what factors would lead you to trust the blood test, or not trust the blood test.

Assume you are a social scientist and wish to study a subject of your choosing (e.g., social, religious, political issues, etc.). State the question you wish to study. Then, form the null and alternative hypotheses. Describe the process you would use to test your hypothesis including the test statistic and the level of significance you would use.

In Roper vs Simmons, the US Supreme Court ruled that juvenile defendants can no longer be sentenced to death in a capital trial. They cited information submitted by the American Psychological Association showing that age is correlated with impulsiveness (young people are often impulsive, a factor that might lead them to commit a homicide). Imagine the Court is unsure whether to categorize 16-, 17-, 18-, or 19-year-olds as adults or as juveniles. However, they do have an impulsiveness threshold. If the defendant’s impulsiveness score (Y') is lower than or equal to an impulsiveness threshold of 50 set by the Court, the defendant will be considered a juvenile and thus will be ineligible for the death penalty. Using X for age (the predictor variable) with a slope (b) of 2 an α of 14, are 19-year-olds eligible for the death penalty? What about 16-, 17-, and 18-year-olds?

Define: Sensitivity, False positives, Specificity, Accuracy, a double-blind trial, an open trial, an interim analysis, and when to use a Fisher exact test or a Chi-square test?

Consider the probability that exactly 95 out of 158 computers will not crash in a day. Assume the probability that a given computer will not crash in a day is 56%. Approximate the probability using the normal distribution.

1. The distribution of heights of adult women is Normally distributed, with a mean of 65 inches and a standard deviation of 3.5 inches.Susan's height has a z-score of negative 0.5 when compared to all adult women in this distribution. What does this z-score tell us about how Susan's height compares to other adult women in terms of height?

1) We are creating a new card game with a new deck. Unlike the normal deck that has 13 ranks (Ace through King) and 4 Suits (hearts, diamonds, spades, and clubs), our deck will be made up of the following.

Each card will have:
i) One rank from 1 to 10.
ii) One of 9 different suits.

Hence, there are 90 cards in the deck with 10 ranks for each of the 9 different suits, and none of the cards will be face cards! So, a card rank 11 would just have an 11 on it. Hence, there is no discussion of "royal" anything since there won't be any cards that are "royalty" like King or Queen, and no face cards!

The game is played by dealing each player 5 cards from the deck. Our goal is to determine which hands would beat other hands using probability. Obviously the hands that are harder to get (i.e. are more rare) should beat hands that are easier to get.a) How many different ways are there to get any 5 card hand?
The number of ways of getting any 5 card hand is  
DO NOT USE ANY COMMAS

b)How many different ways are there to get exactly 1 pair (i.e. 2 cards with the same rank)?
The number of ways of getting exactly 1 pair is
DO NOT USE ANY COMMAS

What is the probability of being dealt exactly 1 pair?
Round your answer to 7 decimal places.


c) How many different ways are there to get exactly 2 pair (i.e. 2 different sets of 2 cards with the same rank)?
The number of ways of getting exactly 2 pair is
DO NOT USE ANY COMMAS

What is the probability of being dealt exactly 2 pair?
Round your answer to 7 decimal places.


d) How many different ways are there to get exactly 3 of a kind (i.e. 3 cards with the same rank)?
The number of ways of getting exactly 3 of a kind is
DO NOT USE ANY COMMAS

What is the probability of being dealt exactly 3 of a kind?
Round your answer to 7 decimal places.


e) How many different ways are there to get exactly 4 of a kind (i.e. 4 cards with the same rank)?
The number of ways of getting exactly 4 of a kind is
DO NOT USE ANY COMMAS

What is the probability of being dealt exactly 4 of a kind?
Round your answer to 7 decimal places.


f) How many different ways are there to get exactly 5 of a kind (i.e. 5 cards with the same rank)?
The number of ways of getting exactly 5 of a kind is  
DO NOT USE ANY COMMAS

What is the probability of being dealt exactly 5 of a kind?
Round your answer to 7 decimal places.


g) How many different ways are there to get a full house (i.e. 3 of a kind and a pair, but not all 5 cards the same rank)?
The number of ways of getting a full house is  
DO NOT USE ANY COMMAS

What is the probability of being dealt a full house?
Round your answer to 7 decimal places.


h) How many different ways are there to get a straight flush (cards go in consecutive order like 4, 5, 6, 7, 8 and all have the same suit. Also, we are assuming there is no wrapping, so you cannot have the ranks be 8, 9, 10, 1, 2)?
The number of ways of getting a straight flush is  
DO NOT USE ANY COMMAS

What is the probability of being dealt a straight flush?
Round your answer to 7 decimal places.


i) How many different ways are there to get a flush (all cards have the same suit, but they don't form a straight)?
Hint: Find all flush hands and then just subtract the number of straight flushes from your calculation above.
The number of ways of getting a flush that is not a straight flush is
DO NOT USE ANY COMMAS

What is the probability of being dealt a flush that is not a straight flush?
Round your answer to 7 decimal places.


j) How many different ways are there to get a straight that is not a straight flush (again, a straight flush has cards that go in consecutive order like 4, 5, 6, 7, 8 and all have the same suit. Also, we are assuming there is no wrapping, so you cannot have the ranks be 8, 9, 10, 1, 2)?
Hint: Find all possible straights and then just subtract the number of straight flushes from your calculation above.
The number of ways of getting a straight that is not a straight flush is  
DO NOT USE ANY COMMAS

What is the probability of being dealt a straight that is not a straight flush?
Round your answer to 7 decimal places.

A researcher wanted to learn whether the urge to smoke cigarettes was associated with the number of times a person exercised in the last week. So the researcher gathered 10 who smoke for the study. Participants were asked to rate their urge to smoke on a scale of 0 (no urge) to 10 (extreme urge), and to give the number of times each had worked out in the last week. Using the data below, an alpha of .05 (two-tailed), use a Pearson correlation to determine the outcome.

n the box below, provide the following information:

Null Hypothesis in sentence form (1 point):

Alternative Hypothesis in sentence form (1 point):
Critical Value(s) (2 points):

Calculations (4 points): Note: the more detail you provide, the more partial credit that I can give you if you make a mistake.

Outcome (determination of significance or not, and what this reflects in everyday language, 2 points)

Please answer all parts

1. Confidence Interval Given. Assume I created a 95% confidence interval for the mean hours studied for a test based on a random sample of 64 students. The lower bound of this interval was 3.1416 and the upper bound was 18.6282. Assume that when I created this interval I knew the population standard deviation. Keep all decimals in your calculations.

a) Calculate the width of the interval.

(b) Calculate the margin of error for the interval.

(c) Calculate the center of the interval.

(d) What is the sample mean?

(e) What is the z ∗ (or zα/2) used? (

f) Calculate the population standard deviation. [Do not use the Empirical Rule.]

Dementia is the loss of the intellectual and social abilities severe enough to interfere with judgment, behavior, and daily functioning. Alzheimer’s disease is the most common type of dementia. In the article “Living with Early Onset Dementia: Exploring the Experience and Developing Evidence-Based Guidelines for Practice” (Alzheimer’s Care Quarterly, Vol. 5, Issue 2, pp. 111–122), P. Harris and J. Keady explored the experience and struggles of people diagnosed with dementia and their families. A simple random sample of 21 people with early-onset dementia gave the following data on age at diagnosis, in years. (60 58 52 58 59 58 51 61 54 59 55 53 44 46 47 42 56 57 49 41 43) At the 1% significance level, do the data provide sufficient evidence to conclude that the mean age at diagnosis of all people with early-onset dementia is less than 55 years old? Assume that the population standard deviation is 6.8 years and note that ¯x = 52.5 years.

This is for an EXCEL formula.

Problem 23. Contrast the probability distribution for the value of the 2 card hand dealt form standard deck of 52 cards. (All Face cards have a value of 10 and the Ace has value of 11).

What is the probability of being dealt 21?

What is the probability of being dealt 16? Construct a chart for the cumulative distribution function.

What is the probability of being a 16 or less?

Between 12 and16?

Between 17 and 20?

Find the expected value and standard deviation of 2 card hand

Similar to the study described on Handout 11, investigators recorded PWV, a measure of vascular stiffness, in 18 children diagnosed with progeria. The objective was to test the effectiveness of the drug lonafarnib. PWV was measured on the 18 children before taking the drug, then re-measured on the same children after receiving a daily dose of the drug for two years. Please see HW3b data, where “untreated” = before taking the drug, and “treated” = after taking the drug for two years.

a. Report the null and alternate hypotheses

b. Check assumptions of the matched pairs T-Test with a dotplot or histogram and report if it is reasonable to conduct the test

c. Run the test and report the appropriate test statistic, df, and P value

d. Based on your results from part c, do you accept or reject the null hypothesis?

e. Report the 95% CI for the mean difference between untreated and treated and interpret what this means.

A production line operation is designed to fill cartons with laundry detergent to a mean weight of 64 ounces. A sample of cartons is periodically selected and weighed to determine whether underfilling or overfilling is occurring. If the sample data lead to a conclusion of underfilling or overfilling, the production line will be shut down and adjusted to obtain proper filling.

(a) Formulate the null and alternative hypotheses that will help in deciding whether to shut down and adjust the production line.

a) H₀: μ = 64

b) H_a: μ ≠ 64

a) H₀: μ ≤ 64

b) H_a: μ > 64

a)H₀: μ = 64

b)H_a: μ < 64

a) H₀: μ = 64

b) H_a: μ > 64

a) H₀: μ ≥ 64

b) H_a: μ < 64

(b) Comment on the conclusion and the decision when H₀ cannot be rejected.

a) Conclude that there is not statistical evidence that the production line is not operating properly. Do not allow the production process to continue.

b) Conclude that there is statistical evidence that the production line is not operating properly. Do not allow the production process to continue.    

c) Conclude that there is statistical evidence that the production line is not operating properly. Allow the production process to continue.

d) Conclude that there is not statistical evidence that the production line is not operating properly. Allow the production process to continue.

(c) Comment on the conclusion and the decision when H₀ can be rejected.

a) Conclude that there is not statistical evidence that overfilling or underfilling exists. Do not shut down and adjust the production line.

b) Conclude that there is statistical evidence that overfilling or underfilling exists. Shut down and adjust the production line.    

c) Conclude that there is not statistical evidence that overfilling or underfilling exists. Shut down and adjust the production line.

d) Conclude that there is statistical evidence that overfilling or underfilling exists. Do not shut down and adjust the production line.

A chocolate chip cookie manufacturing company recorded the number of chocolate chips in a sample of

6060

cookies. The mean is

23.8523.85

and the standard deviation is

2.882.88.

Construct a

8080​%

confidence interval estimate of the standard deviation of the numbers of chocolate chips in all such cookies.

The United States Centers for Disease Control and Prevention (CDC) found that 17.9%17.9% of women ages 1212–5959 test seropositive for HPV‑16. Suppose that Tara, an infectious disease specialist, assays blood serum from a random sample of n=1000n=1000 women in the United States aged 1212–59.59.

Apply the central limit theorem for the distribution of a sample proportion to find the probability that the proportion, ^p,p^, of women in Tara's sample who test positive for HPV‑16 is greater than 0.2010.201. Express the result as a decimal precise to three places.

P(^p>0.201)=P(p^>0.201)=

Apply the central limit theorem for the distribution of a sample proportion to find the probability that the proportion of women in Tara's sample who test positive for HPV‑16 is less than 0.1730.173. Express the result as a decimal precise to three places.

P(^p<0.173)=P(p^<0.173)=