* Make sure you turn in your code (take a screen shot of your code in R)and answers. Conduct the hypothesis
and solve questions by using R.
2) A random sample of 12 graduates of a secretarial school averaged 73.2 words per minute
with a standard deviation of 7.9 words per minute on a typing test. What can we conclude,
at the .05 level, regarding the claim that secretaries at this school average less than 75
words per minute on the typing test? (You may treat the number of words that a secretary
types in one minute as being normally distributed.)

The number of buses that arrive at a bus stop during a one-hour time span can be modeled as a Poisson process with rate λ (see Remark below). Now suppose a passenger has just arrived at the bus stop and starts waiting. Let Y be the time (in unit of hours) she needs to wait to see the first bus.

(a) Is Y a discrete or continuous random variable? Find the set of all possible values of Y .

(b) For a possible value y of Y , find P(Y > y).

(c) Find the distribution (pdf/pmf) of Y and identify it as one of the “named” distributions with corresponding parameter(s).

In the same setting as in the previous question, suppose we know there is exactly one bus arrival on a time interval [0, 1] (the unit is hour). Let Y be this single arrival time.

(d) Find the set of possible values of Y .

(e) For a possible value y of Y , find P(Y ≤ y).

(f) Find the pdf/pmf of Y and identify it as one of the “named” distributions with corresponding parameter(s).

A simple random sample of size n is drawn from a population that is normally distributed. The sample​ mean, x overbar​, is found to be 110​, and the sample standard​ deviation, s, is found to be 10. ​(a) Construct an 80​% confidence interval about mu if the sample​ size, n, is 28. ​(b) Construct an 80​% confidence interval about mu if the sample​ size, n, is 13. ​(c) Construct a 70​% confidence interval about mu if the sample​ size, n, is 28. ​(d) Could we have computed the confidence intervals in parts​ (a)-(c) if the population had not been normally​ distributed?

A person wants to invest $10,000 into stocks: a high tech company (T) with an expected annual return of 12% and a risk index of 8; and a regulated power company (P) with an expected annual return of 6% and a risk index of 2. To limit risk, the combined portfolio risk must be no more than 6 and the proportion of investment in T must be less than 60%. Find the portfolio that will maximize the annual return R while meeting the risk limitations.

a. Formulate the Investment Portfolio problem with the requirement of investing up to $10,000, and solve it graphically.

b. Compute the increase in annual return if the total investment is increased by $1,000.

c. Compute the increase in annual return if the constraint of portfolio risk index is increased from 6 to 7.

Here is a list of scores on the Traditionalism Index for the ANES unique group of

respondents who simultaneously rated Christian Fundamentalists and Atheists

at 100 on the Feeling Thermometer:

7, 10, 5, 6, 5, 8, 5, 12, 4, 11, 11, 9, 10, 13, 8, 8, 7

Use these raw data to the find the mean, median, mode, variance, and standard

deviation.

A trucking company wants to find out if their drivers are still alert after driving long hours. So, they give a test for alertness to two groups of drivers. They give the test to 400 drivers who have just finished driving 4 hours or less and they give the test to 585 drivers who have just finished driving 8 hours or more. The results of the tests are given below.

Passed Failed
Drove 4 hours or less 310 90
Drove 8 hours or more 415 170

Is there is a relationship between hours of driving and alertness? (Do a test for independence.) Test at the 1 % level of significance.

Case studies showed that out of 10,409 convicts who escaped from certain prisons, only 7958 were recaptured.

(a) Let p represent the proportion of all escaped convicts who will eventually be recaptured. Find a point estimate for p. (Round your answer to four decimal places.)


(b) Find a 99% confidence interval for p. (Round your answers to three decimal places.)

A simple random sample of checks were categorized based on the number of cents on the written check and recorded below. Cents Category 0¢-24¢ 25¢-49¢ 50¢-74¢ 75¢-99¢ Frequency 58 37 28 17 Use the p-value method and a 5% significance level to test the claim that 50% of the check population falls into the 0¢-24¢ category, 20% of the check population falls into the 25¢-49¢ category, 16% of the check population falls into the 50¢-74¢ category, and 14% of the check population falls into the 75¢-99¢ category. Calculate the expected value for the 50¢-74¢ category (round to the nearest tenth).

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 0.1990.199. Express the result as a decimal precise to three places.

P(^p>0.199)=

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 0.1740.174. Express the result as a decimal precise to three places.

P(^p<0.174)=

(Please answer this question accuratelly THANKS)

The following commands in R computes 5000 simulations of sample means of size 12 from a normal distribution with mean µ = 100 and standard deviation σ = 14. require

(fastR2) nsamplesum <- do(5000) * c(sample.mean=mean(rnorm(12,100,14)))

The following commands compute the approximate mean and standard deviation of the sample mean and plot the histogram giving the approximate distribution of the sample mean.

mean(∼ sample.mean, data=nsamplesum) sd(∼ sample.mean, data=nsamplesum) gf dhistogram(∼ sample.mean, data= nsamplesum, bins=20)

(a) Compare the approximate values of mean and standard deviation of the sample mean found above with the expected theoretical ones.

(b) Repeat the same simulation as above using now samples from a uniform distribution in the interval [−2, 4]. Also in this case, run a numerical test over 5000 simulations, compute mean and standard deviation of the sample mean, and compare it to the theoretical result.

Compare the kk-NN classifier, linear discriminant analysis (LDA) and the logistic model when it comes to classification. Which is generally better?

26. A sample of 1100 computer chips revealed that 62% of the chips fail in the first 1000 hours of their use. The company's promotional literature states that 60% of the chips fail in the first 1000 hours of their use. The quality control manager wants to test the claim that the actual percentage that fail is different from the stated percentage. State the null and alternative hypotheses.

H0:

Ha:

27. A sample of 1100 computer chips revealed that 62% of the chips fail in the first 1000 hours of their use. The company's promotional literature states that 60% of the chips fail in the first 1000 hours of their use. The quality control manager wants to test the claim that the actual percentage that fail is different from the stated percentage. Make the decision to reject or fail to reject the null hypothesis at the 0.10 level.

An article suggested that yield strength (ksi) for A36 grade steel is normally distributed with μ = 45 and σ = 4.5.

(a) What is the probability that yield strength is at most 40? Greater than 63? (Round your answers to four decimal places.)

(b) What yield strength value separates the strongest 75% from the others? (Round your answer to three decimal places.)
ksi

A study of long-distance phone calls made from the corporate offices of the Pepsi Bottling Group Inc. showed the calls follow the normal distribution. The mean length of time per call was 4.2 minutes and the standard deviation was 0.60 minutes

a) What is the probability the calls lasted between 3.5 and 4.1 minutes?

b) What is the probability the calls lasted less than 3.4 minutes?

c) As part of her report to the president, the director of communications would like to report the minimum length of the longest (in duration) 4% of the calls. What is this time?

d) As part of her report to the president, the director of communications would like to report the maximum length of the shortest (in duration) 8% of the calls. What is this time?

Suppose we wish to test the hypothesis H0 :μ=45vs. H1 :μ>45.

What will be the result if we conclude that the mean is 45 when the actual mean is 50? Choose one of the following.

1. We have made a Type I error.
2. We have made a Type II error.
3. We have made the correct decision.