Work Exercises 1 and 2 using the formula for the probability density function and a hand calculator. Do not use EXCEL. Show all of your work.

1. Assume 45% of all persons three years of age and older wear glasses or contact lenses, For a randomly selected group of seven people, what is the probability that
   1. Exactly 3 wear glasses or contact lenses?
   2. At least 3 wear glasses or contact lenses?
   3. At most 5 wear glasses or contact lenses?
   4. Between 2 and 6 wear glasses or contact lenses?

2. Assume 75% of youths 12-17 years of age have a systolic blood pressure less than 136 mm of mercury. What is the probability that a sample of 12 youths of that age will include
   1. Exactly 8 who have systolic blood pressure less than 136?
   2. Less than 10 who have systolic blood pressure less than 136?
   3. At least 10 who have systolic blood pressure less than 136?
   4. Between 5 and 8 who have systolic blood pressure less than 136?

3. Use EXCEL to complete the entries in the following table. Assume
   1. X represents the number of current cigarette smokers in a sample of size 10.
   2. Each selected subject has a 25% of being a current cigarette smoker.
   3. Being a current cigarette smoker is independent for all subjects

Let P(A) = 0.40, P(B) = 0.20, P(C) = 0.50, P(D) = 0.30, P(A ∩ B) = 0.15, P(A | C) = 0.60, P(B | C) = 0.20, P(B ∩ D) = 0.10, and C and D are mutually exclusive.

Find ...

a. P(C ∩ D)

b. P(C U D)

c. P(B ∩ C)

d. Which one of the following pairs is a pair of statistically independent events? (A and C) (B and D) (B and C) (C and D)

The taxi and takeoff time for commercial jets is a random variable x with a mean of 8.5 minutes and a standard deviation of 3.1 minutes. Assume that the distribution of taxi and takeoff times is approximately normal. You may assume that the jets are lined up on a runway so that one taxies and takes off immediately after the other, and that they take off one at a time on a given runway.

(a) What is the probability that for 34 jets on a given runway, total taxi and takeoff time will be less than 320 minutes? (Round your answer to four decimal places.)


(b) What is the probability that for 34 jets on a given runway, total taxi and takeoff time will be more than 275 minutes? (Round your answer to four decimal places.)


(c) What is the probability that for 34 jets on a given runway, total taxi and takeoff time will be between 275 and 320 minutes? (Round your answer to four decimal places.)

It's true — sand dunes in Colorado rival sand dunes of the Great Sahara Desert! The highest dunes at Great Sand Dunes National Monument can exceed the highest dunes in the Great Sahara, extending over 700 feet in height. However, like all sand dunes, they tend to move around in the wind. This can cause a bit of trouble for temporary structures located near the "escaping" dunes. Roads, parking lots, campgrounds, small buildings, trees, and other vegetation are destroyed when a sand dune moves in and takes over. Such dunes are called "escape dunes" in the sense that they move out of the main body of sand dunes and, by the force of nature (prevailing winds), take over whatever space they choose to occupy. In most cases, dune movement does not occur quickly. An escape dune can take years to relocate itself. Just how fast does an escape dune move? Let x be a random variable representing movement (in feet per year) of such sand dunes (measured from the crest of the dune). Let us assume that x has a normal distribution with μ = 10 feet per year and σ = 2.7 feet per year.

Under the influence of prevailing wind patterns, what is the probability of each of the following? (Round your answers to four decimal places.)

(a) an escape dune will move a total distance of more than 90 feet in 7 years


(b) an escape dune will move a total distance of less than 80 feet in 7 years


(c) an escape dune will move a total distance of between 80 and 90 feet in 7 years

Here is the dataset containing plant growth measurements of plants grown in solutions of commonly-found chemicals in roadway runoff. Researchers wish to determine roadway runoff with different compositions has a different effect on plant growth.

Phragmites australis, a fast-growing non-native grass common to roadsides and disturbed wetlands of Tidewater Virginia, was grown in a greenhouse and watered with one of the following treatments:

• Distilled water (control);
• A weak petroleum solution (representing standard roadway runoff);
• Sodium chloride solution;
• Magnesium chloride solution;
• De-icing brine (50% sodium chloride and 50% magnesium chloride).

Twenty grass preparations were used for each solution, and total growth (in cm) was recorded after watering every other day for 40 days.

1.) Perform the correct statistical test to determine the p-value.

• Report your answer in scientific notation
  • e.g. 1.00E-04

2.) Based on the p-value from your Phragmites dataset analysis, select the options that are TRUE.

3.) Which is / are (an) appropriate evaluation(s) of the results of your Phragmites data analysis?

Let x represent the dollar amount spent on supermarket impulse buying in a 10-minute (unplanned) shopping interval. Based on a certain article, the mean of the x distribution is about $47 and the estimated standard deviation is about $8.

(a) Consider a random sample of n = 80 customers, each of whom has 10 minutes of unplanned shopping time in a supermarket. From the central limit theorem, what can you say about the probability distribution of x, the average amount spent by these customers due to impulse buying? What are the mean and standard deviation of the x distribution?

The sampling distribution of x is approximately normal with mean μ_x = 47 and standard error σ_x = $0.89.The sampling distribution of x is not normal.    The sampling distribution of x is approximately normal with mean μ_x = 47 and standard error σ_x = $0.10.The sampling distribution of x is approximately normal with mean μ_x = 47 and standard error σ_x = $8.

Is it necessary to make any assumption about the x distribution? Explain your answer.

It is necessary to assume that x has an approximately normal distribution.

It is not necessary to make any assumption about the x distribution because μ is large.   

It is necessary to assume that x has a large distribution.

It is not necessary to make any assumption about the x distribution because n is large.

(b) What is the probability that x is between $45 and $49? (Round your answer to four decimal places.)


(c) Let us assume that x has a distribution that is approximately normal. What is the probability that x is between $45 and $49? (Round your answer to four decimal places.)


(d) In part (b), we used x, the average amount spent, computed for 80 customers. In part (c), we used x, the amount spent by only one customer. The answers to parts (b) and (c) are very different. Why would this happen?

The x distribution is approximately normal while the x distribution is not normal.

The standard deviation is larger for the x distribution than it is for the x distribution.    

The standard deviation is smaller for the x distribution than it is for the x distribution.

The sample size is smaller for the x distribution than it is for the x distribution.

The mean is larger for the x distribution than it is for the x distribution.

In this example, x is a much more predictable or reliable statistic than x. Consider that almost all marketing strategies and sales pitches are designed for the average customer and not the individual customer. How does the central limit theorem tell us that the average customer is much more predictable than the individual customer?

The central limit theorem tells us that small sample sizes have small standard deviations on average. Thus, the average customer is more predictable than the individual customer.

The central limit theorem tells us that the standard deviation of the sample mean is much smaller than the population standard deviation. Thus, the average customer is more predictable than the individual customer.    

A company has sales of automobiles in the past three years as given in the table below. Using trend and seasonal components, predict the sales for each quarter of year 4.

Complete all of the steps to derive the normal equations for simple linear regression and then solve them.

A study of the amount of time it takes a mechanic to rebuild the transmission for a 1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their mean rebuild time exceeds 7.7 hours.

A. 0.8531

B. 0.9634

C. 0.9712

D. 0.9931

A sample of 11 individuals shows the following monthly incomes.

PLEASE DON’T COPY PASTE FEOM PREVIOUS QUESTION

1)On Planet Geometry, whenever two right angles have children they can have rectangles or squares with equal probability each. Consider a (very nice) pair of right angles that have 2 children.

a) Using a tree diagram, what is the probability that both children are squares given that at least one is a square? (It is not 1⁄2!)

2) Suppose that P(A∩B)=.3,P(A)=.6, and P(B)=.5.

a. Are A and B mutually exclusive?
b. Are A and B independent?

A recent headline announced / impeachment causes a dip in a President’s Approval Rating.
This conclusion was based on a University poll of 100 adults. In the poll, 41.1% of respondents approved of the president’s job performance.

(a) Based on this poll, what is the probability that more than 50% of the population approve of the president’s job performance?
(b) Form a 90% confidence interval for this estimate.

(c) Assume we know without sampling error the population support was 42% before the impeachment story broke in September. Test whether the impeachment proceedings have actually caused a dip in approval from the 42% baseline at the .05 significance level.

Problem 1 (3 + 3 + 3 = 9) Suppose you draw two cards from a deck of 52 cards without replacement. 1) What’s the probability that both of the cards are hearts? 2) What’s the probability that exactly one of the cards are hearts? 3) What’s the probability that none of the cards are hearts?

Problem 2 (4) A factory produces 100 unit of a certain product and 5 of them are defective. If 3 units are picked at random then what is the probability that none of them are defective?

Problem 3 (3+4=7) There are 3 bags each containing 100 marbles. Bag 1 has 75 red and 25 blue marbles. Bag 2 has 60 red and 40 blue marbles. Bag 3 has 45 red and 55 blue marbles. Now a bag is chosen at random and a marble is also picked at random. 1) What is the probability that the marble is blue? 2) What happens when the first bag is chosen with probability 0.5 and other bags with equal probability each?

Probem 4 (3+3+4=10) Before each class, I either drink a cup of coffee, a cup of tea, or a cup of water. The probability of coffee is 0.7, the probability of tea is 0.2, and the probability of water is 0.1. If I drink coffee, the probability that the lecture ends early is 0.3. If I drink tea, the probability that the lecture ends early is 0.2. If I drink water, the lecture never ends early. 1) What’s the probability that I drink tea and finish the lecture early? 2) What’s the probability that I finish the lecture early? 3) Given the lecture finishes early, what’s the probability I drank coffee?

Problem 5 (4+4+4=12) We roll two fair 6-sided dice. Each one of the 36 possible outcomes is assumed to be equally likely. 1) Find the probability that doubles were rolled. 2) Given that the roll resulted in a sum of 4 or less, find the conditional probability that doubles were rolled. 3) Given that the two dice land on different numbers, find the conditional probability that at least one die is a 1. Problem 6 (8) For any events A, B, and C, prove the following equality: P(B|A) P(C|A) = P(B|A ∩ C) P(C|A ∩ B)

A European growth mutual fund specializes in stocks from the British Isles, continental Europe, and Scandinavia. The fund has over 375 stocks. Let x be a random variable that represents the monthly percentage return for this fund. Suppose x has mean μ = 1.2% and standard deviation σ = 0.5%.

(a) Let's consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all European stocks. Is it reasonable to assume that x (the average monthly return on the 375 stocks in the fund) has a distribution that is approximately normal? Explain. , x is a mean of a sample of n = 375 stocks. By the , the x distribution approximately normal.

(b) After 9 months, what is the probability that the average monthly percentage return x will be between 1% and 2%? (Round your answer to four decimal places.)

(c) After 18 months, what is the probability that the average monthly percentage return x will be between 1% and 2%? (Round your answer to four decimal places.)

(d) Compare your answers to parts (b) and (c). Did the probability increase as n (number of months) increased? Why would this happen?

Yes, probability increases as the mean increases.

Yes, probability increases as the standard deviation decreases.

No, the probability stayed the same.

Yes, probability increases as the standard deviation increases.

(e) If after 18 months the average monthly percentage return x is more than 2%, would that tend to shake your confidence in the statement that μ = 1.2%? If this happened, do you think the European stock market might be heating up? (Round your answer to four decimal places.)

P(x > 2%) = ?????

Explain. This is very likely if μ = 1.2%. One would suspect that the European stock market may be heating up.

This is very likely if μ = 1.2%. One would not suspect that the European stock market may be heating up.

This is very unlikely if μ = 1.2%. One would not suspect that the European stock market may be heating up.

This is very unlikely if μ = 1.2%. One would suspect that the European stock market may be heating up.

You are interested in determining whether an experimental birth control pull has the side effects of changing blood pressure. You plan to randomly sample 18 women from the city in which you live. You give half of them a placebo for a month and then measure their diastolic blood pressure. Then you switch them to the birth control pill for a month and again measure their blood pressure. The other 9 women receive the same treatment except they are given the birth control pill first for a month, followed by the placebo for a month. The blood pressure readings are shown here. What is the power of your study to reject the null assuming that the alternative hypothesis is true. Assume alpha =.05 non directional alternative hypothesis. YOU DO NOT NEED ANY DATA IT ONLY LOOKS FOR POWER

What is the power to detect a false null hypothesis assuming Preal=.9. Put your work and answer below.

What is the probability of failing to reject the null if the null is false assuming Preal =.9. Put your work and answer below.

What is the power to detect a false null hypothesis assuming Preal=.7. Put your work and answer below.

What is the probability of making a type 2 error assuming Preal =.7. Put your work and answer below.