SupposeSuppose the heights of 18-year-old men are approximately normally distributed, with mean 66 inches and standard deviation 1 inches.

(a) What is the probability that an 18-year-old man selected at random is between 65 and 67 inches tall? (Round your answer to four decimal places.)


(b) If a random sample of nine 18-year-old men is selected, what is the probability that the mean height x is between 65 and 67 inches? (Round your answer to four decimal places.)


(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.

The probability in part (b) is much higher because the standard deviation is larger for the x distribution.    

The probability in part (b) is much higher because the mean is smaller for the x distribution.

The probability in part (b) is much higher because the mean is larger for the x distribution.

The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.

Part 1: Probability

1. A question on a multiple-choice test has 10 questions each with 4 possible answers (a,b,c,d).
What is the probability that you guess all the answers to the 10 questions correctly (earn 100%)?

2. If you ask three strangers about their birthdays, what is the probability
a. All were born on Wednesday?
b. What is the probability that all three were born in the same month?
c. All were born on different days of the week?
d. None was born on Saturday?

3. An automobile dealer decides to select a month for its annual sale. Find the probability that it will be September or October.
4. Consider the example of finding the probability of selecting a black card or a 6 from a deck of 52 cards.

5. The probability that s student owns a car is 0.65, and the probability that a student owns a computer is 0.82.

a. If the probability that a student owns both is 0.55, what is the probability that a randomly selected student owns a car or computer?
b. What is the probability that a randomly selected student does not own a car or computer?
6. If 2 cards are selected from a standard deck of cards. The first card is placed back in the deck before the second card is drawn. Find the following probabilities:
a) P(Heart and club) b) P(Red card and 4 of spades) c) P(Spade and Ace of hearts)

7. If 2 cards are selected from a standard deck of cards. The first card is not replaced in the deck before the second card is drawn. Find the following probabilities:
a) P(2 Aces) b) P(Queen of hearts and a King) c) P(Q of Hearts and Q of hearts )
8. A flashlight has 6 batteries, 2 of which are defective. If 2 are selected at random without replacement, find the probability that both are defective.
9. You take a trip by air that involves three independent flights. If there is an 80% chance each specific leg of the trip is on time, what is the probability all three flights arrive on time?

One of the important features of a camera is the battery life as measured by the number of shots taken until the battery needs to be recharged. The data shown in the table below contain the battery life of 10 subcompact and 10 compact cameras. Complete parts​ (a) through​ (c).

. Assuming that the population variances from both types of digital cameras are​ equal, is there evidence of a difference in the mean battery life between the two types of​ cameras? Use

alpha equals 0.05α=0.05.

▼

Do not reject

Reject

Upper H 0H0.

There is

▼

insufficient

sufficient

evidence that the means differ.

b. Determine the​ p-value in​ (a) and interpret its meaning.

​p-value=?

​(Round to two decimal places as​ needed.)

Interpret the​ p-value. Choose the correct answer below.

A.

The probability of obtaining a sample that yields a t test statistic farther away from 0 in the positive direction than the computed test statistic if there is no difference in the mean battery life between the two types of digital cameras.

B.

The probability of obtaining a sample that yields a t test statistic farther away from 0 in the negative direction than the computed test statistic if there is no difference in the mean battery life between the two types of digital cameras.

C.

The probability of obtaining a sample that yields a t test statistic farther away from 0 in either direction than the computed test statistic if there is no difference in the mean battery life between the two types of digital cameras.

c. What is meant by a​ “.05” level of significance​ ?

A.

the lowest level of significance for which we do not reject the null is​ 5%

B.

The probability of a Type II error is​ 5%

C.

we have a​ 5% chance of rejecting the null hypothesis when the null hypothesis is true

D.

we have a​ 5% chance of not rejecting the null hypothesis when the null is false

1. Consider a biased dice, where the probability of rolling a 3 is 4 9 . The dice is rolled 7 times. If X denotes the number of 3’s thrown, then find the binomial distribution for x = 0, 1, . . . 7 and complete the following table (reproducing it in your written solutions). Give your answers to three decimal places.

2. The Maths Students Society (AUMS) decides to conduct small lottery each week to raise money. A participant must pays $2 to enter and chooses three distinct numbers between 1 and 10 (the order does not matter). If their three chosen numbers match the three numbers drawn by AUMS they win a $70 jackpot offered each week, otherwise they recieve nothing. No two entries can use the same numbers.

(a) How many distinct entries can there be?

(b) Write out the probability distribution of returns for one random entry?

(c) A student enters 15 times in one week with different sets of numbers. Determine their probability of winning and the expected return.

(d) A different student enters once each week for 12 weeks. Determine their probability of winning at least once and the expected return.

(e) Suppose 20 entries are made every week for a year. By first calculating the expected amount raised each week, determine how much money AUMS expects to raise in a year?

31. Consider the function f(x) = 3x⁴ − 8x³ + 1.

(a) Find the derivative f 0 (x), and hence the critical points for the function (for this question give both x and y coordinates).

(b) Classify the critical points using the first derivative test to determine if they are local maximum or minimum or neither.

(c) Find any points of inflection for f(x) (give both x and y coordinates).

1. Calculate relative frequencies for; Overall Type of Calls, Call Quality and Call Errors.
2. Calculate the following probabilities:
   • CLM Error and AM Shift
   • COV Error and PM Shift
   • SAV Error or AM Shift
   • Given that the call comes in the morning, what is the probability of a CLM Error
3. NARRATIVE: Based upon the probabilities in step 2, which error(s) should the company focus on for improvement? Should they direct their attention to a particular shift?
4. If a sample size of 15 is used and we assume calls are either correct we can model the call quality using a binomial distribution. Based upon the probability of an error (provided in the Quality Summary under Call Quality), predict the probability of:
   • Exactly 0 errors occurring in 15 calls
   • Less than 2 errors occurring in 15 calls
   • 5 or more errors occurring in 15 calls
5. Provide Descriptive Statistics for Call Time to include; Mean, Standard Deviation and the 5-number summary.
6. Using the Call Time Mean and Standard Deviation from the Quality Sample, find the probability of a call time less than 7 minutes, between 7 and 9 minutes, and greater than 9 minutes.
7. NARRATIVE: From past research the company knows that customers expect to have their calls handled without error and within approximately 7.5 minutes on the phone. Use the information from steps 4-6 to evaluate how well the company is meeting these expectations.

Use the following data on median values of single detached houses of Canadian residents in 20 census metropolitan areas in British Columbia and Ontario in 2017 (source: Statistics Canada) to prepare a statistical report. The data are reported in units of a hundred thousand dollars rounded to the nearest ten thousand dollars (so, for example, 5.7 represents $570,000).

Data: 5.7, 5.2, 12.6, 6.4, 3.7, 2.1, 2.9, 2.4, 4.2, 4.3, 2.9, 3.6, 2.6, 4.2, 4.2, 2.7, 2.5, 2.2, 7.2, 1.9. ):

1. A dotplot or a histogram of the data. Note that you’ll have to group the data into suitable, equal-sized intervals before drawing your graph.

2. A pie graph of the data showing the percentages of the sample in the following categories: 1-3, 3-5, 5-7, 7-10, and 10 or higher.

3. The mean and the median, together with a brief discussion of which of these is the more appropriate measure of what is typical or representative for this dataset.

4. The 5-number summary of the data (i.e., the minimum, lower quartile, median, upper quartile, and maximum

5. The range of the data and the inter-quartile range of the data, together with a brief discussion of exactly what the inter-quartile range represents for this dataset.

6. The following probability calculations, including reasoning. Suppose we select one census metropolitan area at random from the sample of 20. What is the probability that it has a single detached house median value greater than $500k?

7. Suppose we select one census metropolitan area at random from the sample of 20. What is the probability that it has a single detached house median value greater than $500k or less than $200k? (2) Suppose we select two census metropolitan areas at random from the sample of 20. What is the probability that both have a single detached house median value greater than $500k?

1. Objectives:

1) Select a simple random sample by random number table or Excel.

2) Know the sampling distribution of and, and calculate the probabilities by excel.

Q1: The director of personnel for Electronics Associates, Inc (EAI), has been assigned the task of developing a profile of the company’s 250 managers. The characteristics to be identified include the mean annual salary for the managers and the proportion of managers have completed the company’s management training program. Using the 2500 managers as the population for this study. (See data in a file named EAI).

Select a simple random sample of 30 managers from the 2500 EAI managers.

Q2: Business Weej conducted a survey of graduates from 30 top MBA programs (Business-Week, September 22, 2003). On the basis of the survey, assume that the mean annual salary for male and female graduates 10 years after graduation is $168,000 and $117,000, respectively. Assume the standard deviation for the male graduates is $40,000, and for the female graduates, it is $25,000.

a. What is the probability that a simple random sample of 40 male graduates will provide a sample mean within $10,000 of the population mean, $168,000?

b. What is the probability that a simple random sample of 40 female graduates will provide a sample mean within $10,000 of the population mean, $117,000?

c. In which of the preceding two cases, part (a) and part (b), do we have a higher probability of obtaining a sample estimate within $10,000 of the population mean? Comment on the results.

Q3: The Grocery Manufacturers of America reported that 76% of consumers read the ingredients listed on a product’s label. Assume the population proportion p=0.76, and a sample of 400 consumers is selected from the population.

a. Show the sampling distribution of the sample proportion, where is the proportion of the sampled consumers who read the ingredients listed on a product’s label.

b. What is the probability that the sample proportion will be within +- 0.03 of the population proportion?

c. Answer part (b) for a sample of 750 consumers.

The no-show rate for an airline's reservations is 16% in other words the probability is .16 that the person making a reservation will not take the flight. For the next flight, 42 people have reservations. Use normal approximation to the binomial to approximate the following probabilities.

A. what is the probability that exactly 5 people do not make the flight?

B. What is the probability that between 9 and 12 people, inclusive, do not take the flight?

C. What is the probability that at least one person does not take the flight"

D. What is the probability that at most two people do not take the flight?

American Airlines operates flights to a handful of destinations out of Oklahoma City’s Will Rogers World Airport, but let’s think about three in particular: to Dallas, to Charlotte, and to Phoenix. The probability that the Dallas flight is full is 0.6 and assume probabilities of 0.5 and 0.4 for the other cities. Assume the events that the flights to those destinations are independent.

a. What is the probability that all three flights are full?

b. What is the probability that at least one flight is not full?

c. What is the probability that only the Charlotte flight is full?

d. What is the probability that exactly one of the three flights is full?

In the game of roulette, a steel ball is rolled onto a wheel that contains 18 red, 18 black, and 2 green slots. If the ball is rolled 38 times, find the probability of the following events.

A. The ball falls into the green slots 3 or more times.

Probability = (0.3815 is the wrong answer)

B. The ball does not fall into any green slots.

Probability = (0.2590 is wrong)

C. The ball falls into black slots 14 or more times.

Probability = (0.1436 is wrong)

D. The ball falls into red slots 10 or fewer times.

Probability = (0.2967 is the wrong)