part a) x = 137, s = 14.2, n = 20, H0: μ = 132, Ha: μ ≠ 132, α = 0.1

A) Test statistic: t = 1.57. Critical values: t = ±1.645. Do not reject H0. There is not sufficient evidence to conclude that the mean is different from 132.

B) Test statistic: t = 1.57. Critical values: t = ±1.729. Do not reject H0. There is not sufficient evidence to conclude that the mean is different from 132.

C) Test statistic: t = 0.35. Critical values: t = ±1.645. Do not reject H0. There is not sufficient evidence to conclude that the mean is different from 132.

D) Test statistic: t = 0.35. Critical values: t = ±1.729. Do not reject H0. There is not sufficient evidence to conclude that the mean is different from 132.

part b) A local group claims that the police issue more than 60 speeding tickets a day in their area. To prove their point, they randomly select two weeks. Their research yields the number of tickets issued for each day. The data are listed below. At α = 0.01, test the group's claim using P-values.

70 48 41 68 69 55 70 57 60 83 32 60 72 58

A) P-value = 0.4766. Since the P-value is great than α, there is not sufficient evidence to support the the group's claim.

B) P-value = 0.4766. Since the P-value is great than α, there is sufficient evidence to support the the group's claim.

part c) A local school district claims that the number of school days missed by its teachers due to illness is below the national average of μ = 5. A random sample of 28 teachers provided the data below. At α = 0.05, test the district's claim using P-values.

0 3 6 3 3 5 4 1 3 5 7 3 1 2 3 3 2 4 1 6 2 5 2 8 3 1 2 5

A) standardized test statistic ≈ -4.522; Therefore, at a degree of freedom of 27, P must lie between 0.0001 and 0.00003. P < α, reject H0. There is sufficient evidence to support the school district's claim.

B) standardized test statistic ≈ -4.522; Therefore, at a degree of freedom of 27, P must lie between 0.0001 and 0.00003. P < α, reject H0. There is no sufficient evidence to support the school district's claim

part d) To test the effectiveness of a new drug designed to relieve pain, 200 patients were randomly selected and divided into two equal groups. One group of 100 patients was given a pill containing the drug while the other group of 100 was given a placebo. What can we conclude about the effectiveness of the drug if 62 of those actually taking the drug felt a beneficial effect while 41 of the patients taking the placebo felt a beneficial effect? Use α = 0.05.

A) claim: p1 = p2; critical values z0 = ±1.96; standardized test statistic t ≈ 2.971; reject H0; The new drug is effective.

B) claim: p1 = p2; critical values z0 = ±1.96; standardized test statistic t ≈ 2.971; do not reject H0; The new drug is not effective.

Explain the use of the regression model assumptions?

Assume that you have recently purchased a season pass for your favorite football team just like you have been doing for the last 5 years. Suppose that the number of times you expect to go to the games in a season is normally distributed with a mean of 10 games and a standard deviation of 2.4 games. a. What is the probability that you will attend to at least 15 games this season? b. What is the probability that the average number of games you go will be at most 8 in 5 seasons? c. Assume that you think it is worth the money that you paid for the season ticket if you do not miss more than 5% of the games. What is the minimum number of games you need to attend so that you would think the money you paid is worthy?

1. a. Find the z value to the left of mean so that 0.025 percent of the area under the distribution curve lies to the left of it. Also report the area that was left over after the last calculation.

b. Find area between Z= 0 and Z= 1.31

           c. Find probability of P( -1.96≤ z ≤ 1.96) and P (Z=0)                  

2. The average score of a Business Studies graduate is 72 and the standard deviation is 5. The top 2 percent of the graduates receive a certificate of distinction and the next 5 percent are given a chance to apply to the business incubation center for an entrepreneurial venture.

1. What score/marks must you exceed to get a distinction and what range of marks is required for the entrepreneurial opportunity.
2. What average must you get in order to avoid failing.                  

3. The tax rates for basic necessities to be used during the COVID-19 by households items in cents is as follows:

1. Find the 99% and 95% confidence interval for the food tax on the selected items.
2. Which interval is larger and why?
3. If you wanted to convey the true picture of the majority of the people to the government which confidence interval would you report? Give a logical explanation.                                                                                                       

Descriptive Statistics questions MathA100 (Liberal Arts Mathematics)

Gather a sample to analyse, describe how your sample was gathered.

Ask as many people as you can (or at least 30) the following two questions:

"what is the brand of the car you drive?" (Toyota, KIA, Jeep, etc)

"what is the model year of that car?"

1. Construct a bar graph and a pie chart of the brand names

2. Find the mean, median, and standard deviation of the model year

3. Construct a histogram of the model year, decide if it looks approximately normal or not

4. Construct a box plot of the model year, check your data for any outlier

Some common strategies for treating hypertensive patients by nonpharmacologic methods include (1) weight reduction and (2) trying to get the patient to relax more by meditational or other techniques. These strategies were evaluated by randomizing hypertensive patients to four groups who receive the following types of nonpharmacologic therapy: Group 1: patients receive counselling for both weight reduction and meditation. Group 2: patients receive counselling for weight reduction but not for meditation. Group 3: patients receive counselling for meditation but not for weight reduction. Group 4: patients receive no counselling at all. 20 hypertensive patients were assigned at random to each of the four groups, and the change in diastolic blood pressure (DBP) was recorded in these patients after a 1-month period. These data are summarized as follows:
Group n Mean change in DBP    sd of change (baseline - followup) (mm Hg)

1 20 8.6 6.2

2 20 5.3    5.4

3 20 4.9    7.0

4 20 1.1    6.5

(a)Complete the following ANOVA table:
Source of variation    DF Sum of Squares Mean Square    F-statistic p-value

Model (between group) _________ 565.35 _______    _______    < 0.05

Error (Within group)    __________ ________ ____________ __________ _________

Total    3583.31    ___________ _____________    ________    ____________

(b)Interpret this p-value. Be sure to clearly state what is being tested and what this means in the context of the problem of interest. Comment on our ability to draw causal inference in this setting.

Given an array A of n distinct real numbers, we say that a pair of numbers i, j ∈ {0, . . . , n−1} form an inversion of A if i < j and A[i] > A[j]. Let inv(A) = {(i, j) | i < j and A[i] > A[j]}. Answer the following: (a) How small can the number of inversions be? Give an example of an array of length n with the smallest possible number of inversions. (b) Repeat the last exercise with ‘small’ replaced by ‘large’

Imitate the proof of the Master theorem to show that if T(n) ≥ c for all n ≤ n0 and T(n) ≥ aT(n/b) + f(n) where f(n) = Ω(n s ), then (a) if s > logb a, then T(n) = Ω(n s ), (b) if s < logb a, then T(n) = Ω(n logb a ), (c) if s = logb a, then T(n) = Ω(n s log n).

Why is the Mean Square due to Error a better estimate of the population variance than the Mean Square due to Treatment? When is the Mean Square due to Treatment also a good estimate for the population variance? Why?

Problem 1

Suppose that we check for clarity in 50 locations in Lake Tahoe and discover that the average depth of clarity of the lake is 14 feet. Suppose that we know that the standard deviation for the entire lake's depth is 2 feet. What is the confidence interval for clarity of the lake with a 99% confidence level?   

Problem 2 Consider the following exercise: Suppose that a student is taking a multiple-choice exam in which each question has four choices. Assuming that she has no knowledge of the correct answer to any of the questions, she has decided on a strategy in which she will place four balls (marked A, B, C, and D) into a box. She randomly selects one ball for each question and replaces the ball in the box. The marking on the ball will determine her answer to the question.                                                                                                                  

1. If there are five multiple-choice questions on the exam, what is the probability that she will get five questions correct?

2. What is the probability that she will get no more than two questions correct?

      (3) Problem 3    The average number of vehicle arrivals at an intersection is five per minute. Find the probability that thirteen vehicles arrive in 3 minutes.

       ( 4) Problem 4 Researchers have conducted a survey of 1600 coffee drinkers asking how much coffee they drink in order to confirm previous studies. Previous studies have indicated that 72% of Americans drink coffee. The results of previous studies      

                                    Are provided in the survey below.

A classifier is trained on a cancer dataset, and achieves 96% accuracy on new observations. Why might this not be considered a good classifier? How could it be improved?

Questions 3-4: You are guessing at random on an 11-question multiple choice quiz. Each question has five choices, one of which is correct.

3. What is the probability of getting 5 or more questions correct?: *
(A) 0.0117
(B) 0.0504
(C) 0.9496
(D) 0.9883

4. How many questions do you expect to get correct?: *
(A) 2.2
(B) 4.8
(C) 5
(D) 5.5

5. A 95% confidence interval for the proportion of people who believe the Loch Ness Monster exists is (0.6234, 0.7368). What single value would you use to estimate the true proportion of people who believe in the monster?: *
(A) 0.1134
(B) 0.6234
(C) 0.6801
(D) 0.7368

1a) How many five digit number can be written with the digits 1,2,3,4 if there is an even number of twos?

1b) How many five digit number can be written with the digits 1,2,3,4 if two can be used either 4 times, or none?

Q4. In a survey of

3272

adults aged 57 through 85​ years, it was found that

81.1​%

of them used at least one prescription medication.

b. Construct a​ 90% confidence interval estimate of the percentage of adults aged 57 through 85 years who use at least one prescription medication.

nothing​%less than<pless than<nothing​%

​(Round to one decimal place as​ needed.)

What do the results tell us about the proportion of college students who use at least one prescription​ medication?

A.The results tell us​ that, with​ 90% confidence, the probability that a college student uses at least one prescription medication is in the interval found in part​ (b).

B.The results tell us that there is a​ 90% probability that the true proportion of college students who use at least one prescription medication is in the interval found in part​ (b).

C.The results tell us​ that, with​ 90% confidence, the true proportion of college students who use at least one prescription medication is in the interval found in part​ (b).

D.The results tell us nothing about the proportion of college students who use at least one prescription medication.