The CEO of a company wants to estimate the percent of employees that use company computers to go on Facebook during work hours. He selects a random sample of 200 of the employees and finds that 76 of them logged onto Facebook that day. Construct a 95% confidence interval for the population proportion.

Answer the following questions

a. i) Sample proportion = ?    a. ii) critical value Z = ? a. iii) Standard error = ?    a. iv) Margin of error = ?    a. v) Lower limit = ?    a. vi) upper limit = ?

The mean height of an adult giraffe is 19 feet. Suppose that the distribution is normally distributed with standard deviation 0.9 feet. Let X be the height of a randomly selected adult giraffe. Round all answers to 4 decimal places where possible.

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

b. What is the median giraffe height? ft.

c. What is the Z-score for a giraffe that is 21 foot tall?

d. What is the probability that a randomly selected giraffe will be shorter than 19.3 feet tall?

e. What is the probability that a randomly selected giraffe will be between 18 and 18.6 feet tall?

f. The 70th percentile for the height of giraffes is ft.

Suppose that, in an urban population, 50% of people have no exposure to a certain disease, 40% are asymptomatic carriers, and 10% have the disease with symptoms. In a sample of 20 people, we are interested in the probability that the sample contains a number of asymptomatic carriers that is exactly thrice the number that have the disease with symptoms. While the final numerical answer is important, the setup of the solution is moreso.

Given an array A of n distinct 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]}. Define the Inversion problem as follows: • Input: an array A consisting of distinct numbers. • Output: the number of inversions of A, i.e. |inv(A)|. 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’. (c) Show that the running time of insertion sort on array A is Θ(|inv(A)| + n). (d) Assume that the array A contains the number 1, . . . , n in a uniformly random order (among all the n! many orderings). In this case, the running time of insertion sort on A becomes a random variable T. Calculate T up to constant factors.

Define the Exponentiation problem as follows. • Input: A number a and an integer n. • Output: a n . Design an algorithm to solve the problem with as few multiplications as possible

A certain flight arrives on time 89 percent of the time. Suppose 148 flights are randomly selected. Use the normal approximation to the binomial to approximate the probability that

​(a) exactly 134 flights are on time.

​(b) at least 134 flights are on time.

​(c) fewer than 120 flights are on time.

​(d) between 120 and 143​, inclusive are on time.

Shortly after September 11th 2001, a researcher wanted to determine if the proportion of females that favored war with Iraq was significantly different from the proportion of males that favored war with Iraq. In a sample of 73 females, 28 favored war with Iraq. In a sample of 54 males, 29 favored war with Iraq.

a) Let p_F represent the proportion of females that favor the war, p_M represent the proportion of males that favor the war. What are the proper hypotheses?

H₀: p_F = p_M versus H_a: p_F > p_M H₀: p_F = p_M versus H_a: p_F < p_M     H₀: p_F < p_M versus H_a: p_F = p_M H₀: p_F = p_M versus H_a: p_F ≠ p_M

b) What is the test statistic? Compute the statistic using male statistics subtracted from female statistics. Give your answer to four decimal places.

c) What is the P-value for the test? Give your answer to four decimal places.
d) Using a 0.01 level of significance, what conclusion should be reached?

The proportion of females that favor the war and the proportion of males that favor the war are significantly different because the P-value is greater than 0.01. The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is greater than 0.01.      The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is less than 0.01. The proportion of females that favor the war and the proportion of males that favor the war are significantly different because the P-value is less than 0.01.

e) What is the lower endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war? Give your answer to four decimal places.

f) What is the upper endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war? Give your answer to four decimal places

Do male and female skiers differ in their tendency to use a ski helmet? Ruzic and Tudor (2011) report a study in which 710 skiers completed a survey about aspects of their skiing habits. Suppose the results from the question on the survey about ski helmet usage were as follows:

Male helmet usage : Never, ocassionally, Always

244, 58, 192

Female helmet use: 103, 37, 76

Part a)
Which of the following null hypotheses could sensibly be tested by the data presented above?


a. The mean number of male skiers who never wear a ski helmet is the same as the mean number of female skiers who never use one.
b. Male and female skiers are as likely to never use a ski helmet as always use one.
c. There is no relationship between Gender and Helmet usage.

Part b)
Under the null hypothesis, what is the expected number of men in the survey who never wear a ski helmet?

Part c)
Perform a suitable test on the data above to test the null hypothesis.

Almost all medical schools in the United States require students to take the Medical College Admission Test (MCAT). To estimate the mean score ?μ of those who took the MCAT on your campus, you will obtain the scores of an SRS of students. The scores follow a Normal distribution, and from published information you know that the standard deviation is 10.8. Suppose that, unknown to you, the mean score of those taking the MCAT on your campus is 495.

In answering the questions, use z‑scores rounded to two decimal places.

(a) If you choose one student at random, what is the probability that the student's score is between 490 and 500? (Enter your answer rounded to four decimal places.)

(b) You sample 36 students. What is the standard deviation of the sampling distribution of their average score x¯ ? (Enter your answer rounded to two decimal places.)

(c) What is the probability that the mean score of your sample is between 490 and 500? (Enter your answer rounded to four decimal places.)

A manager at a furniture production plant created an incentive plan for her carpenters in order to decrease the number of defects in the furniture production. She wants to check if the incentive plan worked. The manager selected 9 carpenters at random, recorded their annual defects before and after the incentive and came up with the following:

Notice, that a positive outcome of an incentive plan is confirmed with a positive mean of the differences (difference equals before minus after).

Given that the null hypothesis and the alternative hypothesis are:

  H₀: μ_d ≤ 0
  H₁: μ_d > 0

and using a 0.1 significance level, answer the following:

b) Compute the mean of the difference.
For full marks your answer should be accurate to at least two decimal places.

Mean: 0

c) What is the value of the test statistic?
For full marks your answer should be accurate to at least two decimal places.

Test statistic: 0

3. Are younger drivers more likely to have accidents in their driveways? Traffic engineers tabulated types of car accidents by drivers of various ages. Out of a total of 82,486 accidents involving drivers aged 15-24 years, 4243 of them occurred in a driveway. Out of a total of 219,170 accidents involving drivers aged 25-64 years, 10,701 of them occurred in a driveway. Use the 0.05 level of significance.

According to a global Nielson report, 66% of North Americans pay more attention to the nutritional information on a package than they did two years ago. A random sample of 400 individuals from North America showed that 300 people stated that they did pay more attention to the nutritional information on a package than they did two years ago. We are interested in testing to see if the proportion has changed. Calculate the p-value and assess the strength of evidence against the null hypothesis.

A.

p-value = 0.9993, no evidence against the alternate hypothesis

B.

p-value = 0.9993, no evidence against the null hypothesis

C.

p-value = 0.0001, extremely strong evidence against the null hypothesis

D.

p-value = 0.0001, extremely strong evidence against the alternate hypothesis

Independent simple random samples from two strains of mice used in an experiment yielded the following measurements on plasma glucose levels following a traumatic experience: Strain A: 54; 99; 105; 46; 70; 87; 55; 58; 139; 91 Strain B: 93; 91; 93; 150; 80; 104; 128; 83; 88; 95; 94; 97 Do these data provide sufficient evidence to indicate that the variance is larger in the population of strain A mice than in the population of strain B mice? Let a= 0.05. What assumptions are necessary?

Let x be a random variable that represents red blood cell count (RBC) in millions of cells per cubic millimeter of whole blood. Then x has a distribution that is approximately normal. For the population of healthy female adults, suppose the mean of the x distribution is about 4.74. Suppose that a female patient has taken six laboratory blood tests over the past several months and that the RBC count data sent to the patient's doctor are as follows.

(i) Use a calculator with sample mean and standard deviation keys to find x and s. (Round your answers to two decimal places.)

(ii) Do the given data indicate that the population mean RBC count for this patient is lower than 4.74? Use α = 0.05.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: μ < 4.74; H₁: μ = 4.74

H₀: μ = 4.74; H₁: μ < 4.74

    H₀: μ = 4.74; H₁: μ > 4.74

H₀: μ = 4.74; H₁: μ ≠ 4.74

H₀: μ > 4.74; H₁: μ = 4.74

(b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution.

The Student's t, since we assume that x has a normal distribution and σ is known.

The Student's t, since we assume that x has a normal distribution and σ is unknown.

     The standard normal, since we assume that x has a normal distribution and σ is unknown.

The standard normal, since we assume that x has a normal distribution and σ is known.

What is the value of the sample test statistic? (Round your answer to three decimal places.)


(c) Estimate the P-value.

P-value > 0.250

0.100 < P-value < 0.250

     0.050 < P-value < 0.100

0.010 < P-value < 0.050

P-value < 0.010

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

    At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.05 level to conclude that the population mean RBC count for the patient is lower than 4.74.

There is insufficient evidence at the 0.05 level to conclude that the population mean RBC count for the patient is lower than 4.74

Nacirema Airlines is buying a fleet of new fuel-efficient planes. The HogJet and the LitheJet both meet their price and performance needs, and both planes meet EPA noise guidelines. However, the quieter plane is preferred. Each plane is flown through a typical takeoff and landing sequence 10 times, while remote sensors at ground level record the noise levels (in decibels).
The average noise levels from these 10 flights are 80.34 for LitheJet and 82.47 for HogJet, the pooled variance is reported to be equal to 2.73.
Is there enough evidence to conclude that LitheJet planes are quieter?

Show how to calculate test statistics, p val, etc for conclusion.