2. One‐Sample Univariate Hypothesis Testing with Proportions

For this question, show the results “by hand”, but you can use R to check your work. Suppose that the 4‐year graduation rate at a large, public university is 70 percent (this is the population proportion of successes). In an effort to increase graduation rates, the university randomly selected 200 incoming freshman to participate in a peer‐advising program. After 4 years, 154 of these students graduated. What are the null and alternative hypotheses? Can you conclude that this program was a success at the 5‐percent level of significance? Can you conclude that the program is a success at the 1‐percent level of significance? Show your work and explain. Since “success” is an increase in graduation rates, this is a one‐tailed test.

If X ~ N(108, 22.2), calculate the following:

a. P(X = 100, 108, 109, or 11) = _____.

If Y ~ N(-33, 10), then what is the distribution of (Y - (-33))/10?

b. (Y - (-33)/10 ~ N( _____ , _____ )

c. If X1 ~ N(111, 1123), then P(-122 < X1 < 765) = ________.

d. The distribution of (X1 - 111)/1123 is N( ____ , ____ ).

e. P(X1 >= 0) = _______.

f. P(X1 = 0) = _________.

If a doughnut shop offers seven different varieties of doughnut, how many different dozens of doughnuts can one order? Explain your answer.

The amount of time each week that Dunder Mifflin employees spend in pointless meetings follows a normal distribution with a mean of 90 minutes and standard deviation of 10 minutes.

A. What is the probability that employees spend between 81.5 and 103.5 minutes in meetings in a week?

B. The probability that employees spend between 80 and ???? minutes in meetings in a week is equal to 0.8351.

A pair of dice are rolled​ 1,000 times with the following frequencies of​ outcomes:

Use these frequencies to calculate the approximate empirical probabilities and odds for the events a. The sum is less than 3 or greater than 9.

b. The sum is even or exactly divisible by 5.

a. Probabilityequals = ___???

​(Type a​ decimal.)

Odds for = ____??

​(Type a fraction. Simplify your​ answer.)

b. Probabilityequals = ___???

​(Type a​ decimal.)

Odds for = ____??

​(Type a fraction. Simplify your​ answer.)

Select any data set . Use the method of Sections 6-6 to construct a histogram and normal quartile plot, then determine whether the data set appears to come from a normally distributed population.

When one company (A) buys another company(B), some workers of company B are terminated. Terminated workers get severance pay. To be fair, company A fixes the severance payment to company B workers as equivalent to company A workers who were terminated in the last one year. A 36-year-old Mohammed, worked for company B for the last 10 years earning 32000 per year, was terminated with a severance pay of 5 weeks of salary. Bill smith complained that this is unfair that someone with the same credentials worked in company A received more. You are called in to settle the dispute. You are told that severance is determined by three factors; age, length of service with the company and the pay. You have randomly taken a sample of 40 employees of company A terminated last year. You recorded

Number of weeks of severance pay

Age of employee

Number of years with the company

Annual pay in 1000s

Identify best subsets of variables based on Mallows Cp. What is the value of R-square to this “best” model? How many outliers are in the dataset? Use the criteria of your choice and mention it(them)

12. An agricultural research company has developed two new types of soy bean seeds, call them "seed A" and "seed B". A study is conducted to determine which will produce a higher mean yield. To test the two types of seed, 20 similar plots of land were randomly placed into one of two groups. One group of ten plots was planted with "seed A", while the other ten plots were planted with "seed B". The yield of each field, in bushels per acre, was recorded in the table below.

1. Conduct a hypothesis test at a 0.050.05 level of significance to determine if the two types of soy beans produce different mean yields.

The test statistic is ________________________
The p-value is____________________________

Construct a 9595% confidence interval for the mean of the differences. Hint: with the data in your lists, use the two-independent sample t-INTERVAL option on your calculator.
__________________to________________________

13. A company owns 9 trucks of various makes and models. The manager recently heard that inflating tires with nitrogen may provide slightly better gas mileage. The manager wants to determine if there is a noticeable increase in the mean gas mileage for the 9 trucks when nitrogen is utilized. Over a period of time, a test is run in which the gas mileage of each truck is recorded both with and without nitrogen in the tires. The gas mileages of the 15 trucks with and without nitrogen in the tires are recorded here. (data is in miles per gallon)

(b) The test statistic is_____________________

(c) The p-value is________________________

14. A professor of nursing wonders if the female nursing students are more likely to drop out of a nursing program than the male nursing students. To check her intuition, several nursing programs are compiled and random samples of both male and female nursing students are selected. Of the 200 male nursing students selected, 17 of them did not attain their nursing degree. Of the 700 female nursing students selected, 68 of them did not attain their nursing degree. Test the claim that the proportion of females not completing their degree is higher than the proportion of males using a level of significance of 0.05.
The test statistic is _________________
The p-value is ____________________

A point estimator is a sample statistic that provides a point estimate of a population parameter. Complete the following statements about point estimators.

Given two unbiased estimators of the same population parameter, the estimator with thelarger expected value   is consistent   .

A point estimator is said to bebiased   if itsexpected value   is equal to the value of the population parameter that it estimates.

A point estimator is said to beunbiased   if, as the sample size is     , the estimator tends to provide better estimates of the population parameter.

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- Using R Randomization Test -

"When waiting to get someone's parking space, have you ever thought that the driver you are waiting for is taking longer than necessary? Ruback and Juieng (1997) ran a simple experiment to examine that question. They hung out in parking lots and recorded the time that it took for a car to leave a parking place. They broke the data down on the basis of whether or not someone in another car was waiting for the space.

The data are positively skewed, because a driver can safely leave a space only so quickly, but, as we all know, they can sometimes take a very long time. But because the data are skewed, we might feel distinctly uncomfortable using a parametric t test. So we will adopt a randomization test."

```{r}
# no waiting records the time it took a driver to leave the parking spot if no one was waiting for the driver
no_waiting <- c(36.30, 42.07, 39.97, 39.33, 33.76, 33.91, 39.65, 84.92, 40.70, 39.65,
39.48, 35.38, 75.07, 36.46, 38.73, 33.88, 34.39, 60.52, 53.63, 50.62)

# waiting records the time it takes a driver to leave if someone was waiting on the driver
waiting <- c(49.48, 43.30, 85.97, 46.92, 49.18, 79.30, 47.35, 46.52, 59.68, 42.89,
49.29, 68.69, 41.61, 46.81, 43.75, 46.55, 42.33, 71.48, 78.95, 42.06)

mean(waiting)
mean(no_waiting)
obs_dif <- mean(waiting) - mean(no_waiting)
```

Conduct a randomization test to test the hypothesis that there is no difference in average time for drivers who have a person waiting vs those who do not have a person waiting, against the alternative that drivers who have a person waiting will take *longer* than if they did not.

Be sure to calculate an empirical p-value and make the appropriate conclusion.

In the manufacturing process of carbon composition resistors historical records indicate that 80% of these components have superior quality, 10% have very good quality, 5% have marginal quality and 5% are poor (non-shippable). A quality control firm implemented changes to the manufacturing plant (like installing dehumidifiers) and several months later a random sample of 1200 carbon composition resistors we gathered and rated. Here’s what was found:

Superior:996 Very good:126 Marginal:48 Poor:30 Total:1200

Answer the following questions below.
(a) Test if there is evidence to suggest that the implemented change changed the distribution. Use α = .05.

(b) Looking at each of the (Xi−Ei)2 , which is the largest and construct a 95% confidence interval the multi- Ei

nomial proportion corresponding this ratio.
(c) Would you believe that the installation of dehumidifiers improved the quality? Briefly explain.

The numbers below represent heights (in feet) of 3-year old elm trees.

5.1, 5.5, 5.8, 6.1, 6.2, 6.4, 6.7, 6.8, 6.9, 7.0,

7.2, 7.3, 7.3, 7.4, 7.5, 7.7, 7.9, 8.1, 8.1, 8.2,

8.3, 8.5, 8.6, 8.6, 8.7, 8.7, 8.9, 8.9, 9.0, 9.1,

9.3, 9.4, 9.6, 9.8, 10.0, 10.2, 10.2

Using the chi-square goodness-of-fit test, determine whether the heights of 3-year old elm trees are normally distributed, at the a = .05 significance level. Also, find the p- value.

Sheldon Pikyeeter has peculiar eating habits. For dinner, he always eats spaghetti 3 times per week, cheese pizza twice a week, Thai chicken one time per week, and tacos once a week. Note: for all questions below, consider each week as starting on Sunday and ending Saturday!

a) For a particular week, Sheldon is going to list all the possible different dinner plans. (Sunday through Saturday). How many weekly dinner plans are on Sheldon’s list?

b) What is the probability that a weekly dinner plan has spaghetti on Monday and pizza on Friday?

c) What is the probability that a weekly dinner plan has taco night before all of the spaghetti nights?

d) What is the probability that all of the spaghetti nights are on consecutive nights in a week?

e) What is the probability that none of the spaghetti dinners are on consecutive nights?

A regression Study involving 32 convenience stores was undertaken to examine the relationship between monthly newspaper advertising expenditures (X) and the number of the customers shopping at the store (Y). A partial ANOVA table is below.

Complete the mission parts of the table.

Test whether or not X and Y are linearly related using the correlation coefficient. Use alpha = .01

What proportion of the variation in the number of customers is left UNEXPLAINED by this model?

At the 1% level of significance, what is the critical value to test the explanatory power of the model?

Suppose that the miles-per-gallon (mpg) rating of passenger cars is normally distributed with a mean and a standard deviation of 30.9 and 2.7 mpg, respectively. [You may find it useful to reference the z table.]

a. What is the probability that a randomly selected passenger car gets more than 32 mpg? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

b. What is the probability that the average mpg of four randomly selected passenger cars is more than 32 mpg? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)

c. If four passenger cars are randomly selected, what is the probability that all of the passenger cars get more than 32 mpg? (Round “z” value to 2 decimal places, and final answer to 4 decimal places.)