A survey of adults ages​ 18-29 found that 83​%

use the Internet. You randomly select 108 adults ages​ 18-29 and ask them if they use the Internet.

​(a) Find the probability that exactly 86 people say they use the Internet.

​(b) Find the probability that at least 86 people say they use the Internet.

​(c) Find the probability that fewer than 86 people say they use the Internet.

​(d) Are any of the probabilities in parts​ (a)-(c) unusual? Explain.

For the geometric distribution:

1a) Determine the most powerful critical region for testing H₀ p=p₀ against H₁ p=θp (p₁ > p₀) using a random sample of size n.

1b) Find the uniformly most powerful H₀ p<θ₀ against H₁ p>θ₁

The Denver Post reported that, on average, a large shopping center had an incident of shoplifting caught by security 1.2 times every four hours. The shopping center is open from 10 A.M. to 9 P.M. (11 hours). Let r be the number of shoplifting incidents caught by security in an 11-hour period during which the center is open.

(a) Explain why the Poisson probability distribution would be a good choice for the random variable r.

1) Frequency of shoplifting is a rare occurrence. It is reasonable to assume the events are dependent.

2) Frequency of shoplifting is a common occurrence. It is reasonable to assume the events are dependent.   

3) Frequency of shoplifting is a rare occurrence. It is reasonable to assume the events are independent.

4) Frequency of shoplifting is a common occurrence. It is reasonable to assume the events are independent.

What is λ? (Use 2 decimal places.)


(b) What is the probability that from 10 A.M. to 9 P.M. there will be at least one shoplifting incident caught by security? (Use 4 decimal places.)


(c) What is the probability that from 10 A.M. to 9 P.M. there will be at least three shoplifting incidents caught by security? (Use 4 decimal places.)


(d) What is the probability that from 10 A.M. to 9 P.M. there will be no shoplifting incidents caught by security? (Use 4 decimal places.)

The time until failure for an electronic switch has an exponential distribution with an average time to failure of 4 years, so that λ =

= 0.25. (Round your answers to four decimal places.)

(a)

What is the probability that this type of switch fails before year 3?

(b)

What is the probability that this type of switch will fail after 5 years?

(c)

If two such switches are used in an appliance, what is the probability that neither switch fails before year 7?

a professor has learned that six students in her class of 19 will cheat on the exam. She decides to focus her attention on eight randomly chosen students during the exam. A. what is the probability that she finds at least one the students cheating? B. what is the probability that she finds at least one of the students cheating if she focuses on nine randomly chosen students?

Ten professional golfers are asked to hit each of two brands of golf ball with their drivers. The golfers do not know which brand they are hitting, and the order in which they hit the balls is determined by flipping a coin. Is there evidence that Brand 1 golf balls are hit a smaller distance, on average, than Brand 2 golf balls?

Golfer Brand 1 Brand 2

1 265 268

2 281 283

3 260 257

4 274 277

5 269 270

6 288 291

7 271 275

8 270 274

9 267 269

10 284 288

a. No since the p-value is greater than 0.05.

b. Yes at a 0.10 level of significance, but not at the 0.05 level.

c. Yes, since the p-value is less than 0.01.

d. Yes at the 0.05 level of significance, but not the 0.01 level.

Find the following probabilities based on the standard normal variable Z. (You may find it useful to reference the z table. Leave no cells blank - be certain to enter "0" wherever required. Round your answers to 4 decimal places.)

Life Span of Tires: A certain brand of automobile tires has a mean life span of 35,000 miles and a standard deviation or of 2,250 miles. (Assume a bell-shape distribution).

1. The span of three randomly selected tires are 34,000 miles, 37,000 miles, and 31,000 miles. Find the z-scores that correspond to each life span. Would any of these tires be considered unusual?
2. The life spans of three randomly selected tires are 30,500 miles, 37,250 miles, and 35,000 miles. Using the Empirical Rule, find the percentile that corresponds to each life span.

In 2012, the percent of American adults who owned cell phones and used their cell phones to send or receive text messages was at an all-time high of 80%. Assume that 80% refers to the population parameter. More recently in 2015, a polling firm contacts a simple random sample of 110 people chosen from the population of cell phone owners to confirm the percent who use their phone to text. The firm askes each person "do you use your cell phone to send or receive texts Yes or No. "

a) Verify that the conditions are met so that the central limit theorem can apply to p̂

b) What is the approximate distribution of p̂, the proportion of cell phone owners in the 2015 sample who use their cell phone to text? Give the shape, mean, and standard deviation.

c) What is the probability that p̂ is between 78% and 82%: what is P(0.78 < p̂ < 0.82). In other words, what is the probability that p̂ estimates π within 2% of 0.8?

d) Suppose the polling firm increased the number of people in its sample to 1100 people. Now what is the probability that p̂ is between 78% and 82%? In other words, what is the probability that p̂ estimates π within 2%?

e) Which sample size (110 or 1100) gives a more accurate estimate of the population proportion of cell users who text?

A food safety guideline is that the mercury in fish should be below 1 part per million​ (ppm). Listed below are the amounts of mercury​ (ppm) found in tuna sushi sampled at different stores in a major city. Construct a 98​% confidence interval estimate of the mean amount of mercury in the population. Does it appear that there is too much mercury in tuna​ sushi?

0.55  0.71 0.11 0.92  1.34  0.53  0.93 What is the confidence interval estimate of the population mean μ​?

__ppm<μ​<__ppm

1. Determine a correlation matrix for the above variables.
2. Indicate if any collinearity exists between or among the independent variable.
3. Indicate and explain which independent variable gets into the equation first to analyze, second to analyze, etc.. (Keep in mind that the correlation coefficient can take on a value between +/- 1. A negative value simply means a negative relationship between the dependent and independent variables. A positive value implies a positive relationship. A positive sign does not imply it is more important than a negative sign.)
4. Indicate the significance of those independent variables using the p-Value of the slope at a 5% level of significance.
5. Develop the best regression equation, linear or multiple, that would be useful in predicting the dependent variable. Explain your rationale for the equation that you developed.

An oil exploration company purchases drill bits that have a life span that is approximately normally distributed with a mean equal to 88 hours and a standard deviation equal to 12 hours. (Round your answers to four decimal places.)

(a)

What proportion of the company's drill bits will fail before 79 hours of use?

(b)

What proportion will last at least 79 hours?

(c)

What proportion will have to be replaced after more than 97 hours of use?

Construct the confidence interval for the population standard deviation for the given values. Round your answers to one decimal place.

n=5, s=4.3, and c=0.95

Assume the average price for a movie is ​$10.16. Assume the population standard deviation is ​$0.49 and that a sample of 32 theaters was randomly selected. Complete parts a through d below.

a. Calculate the standard error of the mean.

b. What is the probability that the sample mean will be less than 10.31​?

c. What is the probability that the sample mean will be less than $10.11​?

d. What is the probability that the sample mean will be more than ​$10.26​?

(Round everything to four decimal places as needed.)

6. According to Mars Inc., the distribution of colors of the M&M candies is given in the table below. Several bags were randomly chosen and the percentages of each color were found to be as follows: Brown – 15.3%, Yellow – 18.1%, Red – 11.7%, Blue – 19.8%, Orange – 22.6%, and Green – 12.5%. According to this sample, does it appear that the distribution of colors does not fit what’s expected, according to the manufacturer’s percentages? State your hypotheses, test statistic, p-value, and conclusion (in terms of the problem).

Color Percentage

Brown 13%

Yellow 14%

Red 13%

Blue 24%

Orange 20%

Green 16%