Isabel Briggs Myers was a pioneer in the study of personality types. The personality types are broadly defined according to four main preferences. Do married couples choose similar or different personality types in their mates? The following data give an indication.

Suppose that a married couple is selected at random.

1(a) Use the data to estimate the probability that they will have 0, 1, 2, 3, or 4 personality preferences in common. (For each answer, enter a number. Enter your answers to 2 decimal places.)

1(b)  Do the probabilities add up to 1? Why should they?

a) Yes, because they do not cover the entire sample space.

b) No, because they do not cover the entire sample space.     

c) Yes, because they cover the entire sample space.

d) No, because they cover the entire sample space.

What is the sample space in this problem?

a) 0, 1, 2, 3 personality preferences in common

b) 1, 2, 3, 4 personality preferences in common     

c) 0, 1, 2, 3, 4, 5 personality preferences in common

d) 0, 1, 2, 3, 4 personality preferences in common

(2). A particular lake is known to be one of the best places to catch a certain type of fish. In this table, x = number of fish caught in a 6-hour period. The percentage data are the percentages of fishermen who caught x fish in a 6-hour period while fishing from shore.

(a) Convert the percentages to probabilities and make a histogram of the probability distribution. (Select the correct graph.)

2. (b) Find the probability that a fisherman selected at random fishing from shore catches one or more fish in a 6-hour period. (Enter a number. Round your answer to two decimal places.) =______

2(c) Find the probability that a fisherman selected at random fishing from shore catches two or more fish in a 6-hour period. (Enter a number. Round your answer to two decimal places.) =_____

(d) Compute μ, the expected value of the number of fish caught per fisherman in a 6-hour period (round 4 or more to 4). (Enter a number. Round your answer to two decimal places.)
μ = ____fish

2(e) Compute σ, the standard deviation of the number of fish caught per fisherman in a 6-hour period (round 4 or more to 4). (Enter a number. Round your answer to three decimal places.)
σ = _____ fish

Question 12 pts

Which of the following would be a discrete variable?

Group of answer choices

Number of cans of dog food eaten

Number of inches reflecting someone’s height

Amount of soup in a can

Count of students in class

Flag this Question

Question 22 pts

A main characteristic of a discrete probability distribution is that ___________.

Group of answer choices

each probability is between -1 and +1

each probability is between 1 and 2

the probabilities add to 0

the probabilities add to 1

Flag this Question

Question 32 pts

If Sam can purchase up to 3 ice cream cones and the probabilities that he/she purchases 0, 1, 2, or 3, are 0.20, 0.45, 0.35, and 0.05 respectively, is this a valid probability distribution?

Group of answer choices

No, as they do not add to 1

Yes, as they are all positive

No, as no probability is between 0 and 1

Yes, as all are between 0 and 1

Flag this Question

Question 42 pts

If Alex can purchase up to 2 ice cream cones and the probabilities that he/she purchases 0, 1, or 2 are 0.10, 0.50, and 0.40, what is the mean number of cones that he/she will purchase?

Group of answer choices

0.7

1.3

1.0

1.5

Flag this Question

Question 52 pts

When conducting a binomial experiment, how many times will the experiment be repeated?

Group of answer choices

n

2

infinite

p(x)

Flag this Question

Question 62 pts

The mean of a binomial distribution is found using which of the following formulas?

Group of answer choices

p*q

n*p

q*x

p*x

Flag this Question

Question 72 pts

In binomial probabilities, q represents _______.

Group of answer choices

the negative probability

p-1

the probability of x

1-p

Flag this Question

Question 82 pts

A right skewed distribution has most of the data points to the _______ of the distribution.

Group of answer choices

left

right

top

middle

Flag this Question

Question 92 pts

A left skewed distribution can be described as __________.

Group of answer choices

having most data in higher values and a tail of data to the left

having most data in the lower values and a tail of the data to the right

a symmetrical distribution

having all data to the left side of the graph

Flag this Question

Question 102 pts

When considering if a probability distribution follows the expected distribution, we are considering the distribution’s __________.

Group of answer choices

shape

symmetry

normality

goodness of fit

One needs to be careful when writing formulas for simulation.   "=round( rand() * 6, 0)"  will take a value uniformly distributed between 0 and 1, multiply it by 6, and round it to the nearest integer. Using that formula, what is the probability of getting 6?  (Note: probability is a number between 0 and 1; give your answer to three decimal places.)

Hint: rand() is uniformly distributed from 0 to 1. Hence rand()*6 is uniformly distributed from 0 to 6. Only values above 5.5 will be rounded to 6. We can find the probability from this.

Alternatively, we can simulate a thousand values of "=round(rand()*6,0)", say in the range A1:A1000. Now "=COUNTIF (A1:A1000, 6)" will give us a count of the value 6 in the range A1:A1000. From this we can estimate the probability.

Suppose that 30% of all homeowners in an earthquake-prone area of California are insured against earthquake damage. Four homeowners are selected at random; let x denote the number among the four who have earthquake insurance.

(a) Find the probability distribution of x. (Hint: Let S denote a homeowner who has insurance and F one who does not. Then one possible outcome is SFSS, with probability (.3)(.7)(.3)(.3) and associated x value of 3. There are 15 other outcomes.)

(b) What is the most likely value of x?

0

1

1 and 0

3

4

(c) What is the probability that at least two of the four selected homeowners have earthquake insurance?
P (at least 2 of the 4 have earthquake insurance) = ______

Complete parts​ (a) and​ (b) below.

The number of dogs per household in a small town

(a) Find the​ mean, variance, and standard deviation of the probability distribution.

Find the mean of the probability distribution.

u=___(Round to one decimal place if needed)

Find the variance of the probability distribution.

o2=___(round to one decimal place if needed)

find the standard deviation of the probability distribution

o=____(round to one decimal place when needed)

(b) Interpret the results in the context of the​ real-life situation.

A) a household on average has 0.5 dogs with a standard deviation of 0.9 dogs

B. A household on average has

0.5

dog with a standard deviation of

eleven

dogs.

C. A household on average has

0.9

dog with a standard deviation of

0.5

dog.

D.

A household on average has

0.8

dog with a standard deviation of

0.9

dog.

.9

Problem 12-14 (Algorithmic) The management of Madeira Manufacturing Company is considering the introduction of a new product. The fixed cost to begin the production of the product is $30,000. The variable cost for the product is uniformly distributed between $16 and $24 per unit. The product will sell for $50 per unit. Demand for the product is best described by a normal probability distribution with a mean of 1,200 units and a standard deviation of 300 units. Develop an Excel worksheet simulation for this problem. Use 500 simulation trials to answer the following questions: What is the mean profit for the simulation? Round your answer to the nearest dollar. Mean profit = $ What is the probability that the project will result in a loss? Recalculate the numerical value of probability in percent and then round your answer to the nearest whole number. Probability of Loss = % What is your recommendation concerning the introduction of the product? The input in the box below will not be graded, but may be reviewed and considered by your instructor.

1. Three fair dices were rolled.
(a) How many possible outcomes there will be, if the number in each dice was recorded and the order of dices are considered.
(b) How many possible outcomes there will be, if the sum of the dices are recorded.
(c) What is the probability of getting a result with the sum of the three dices exactly equals to 6?
(d) What is the probability of getting a result with the sum of the three dices less than 6?
2. Seven fair coins were flipped and there outcomes of each coin (head or tail) were recorded.
(a) How many possible outcomes there will be, if the order of coins are considered?
(b) How many possible outcomes with exactly 3 heads(and 4 tails)?
(c) What is the probability of getting a result with exactly 3 heads (and 4 tails)?
(d) What is the probability of getting a result with less than 2 head(and more than 5 tails)?

USA Today reported that Parkfield, California, is dubbed the world’s earthquake capital because it sits on top of the notorious San Andreas fault. Since 1857, Parkfield has had a major earthquake on average of once every 22 years.

a) Explain why the Poisson distribution would be a good choice for r = the number of earthquakes in a given time interval.

b) Compute the probability of at least one major earthquake in the next 22 years. Round lambda to the nearest hundredth, and use a calculator.

c) Compute the probability that there will be no major earthquake in the next 22 years. Round lambda to the nearest hundredth, and use a calculator.

d) Compute the probability of at least one major earthquake in the next 50 years. Round lambda to the nearest hundredth, and use a calculator.

e) Compute the probability that there will be no major earthquake in the next 50 years. Round lambda to the nearest hundredth, and use a calculator.

1. The U.S. Department of Transportation reported that during November, 83.4% of Southwest
Airlines’ flights, 75.1% of US Airways’ flights, and 70.1% of JetBlue’s flights arrived on time (USA
Today, January 4, 2007). Assume that this on-time performance is applicable for flights arriving at
concourse A of the Rochester International Airport, and that 40% of the arrivals at concourse Aare
Southwest Airlines flights, 35% are US Airways flights, and 25% are JetBlue flights.
a. An announcement has just been made that Flight 1424 will be arriving at gate in concourse A.
What is the most likely airline for this arrival?
c. What is the probability that Flight 1424 will arrive on time?
d. Suppose that an announcement is made saying that Flight 1424 will be arriving late. What is the
most likely airline for this arrival? What is the least likely airline?

2. In San Francisco, 30% of workers take public transportation daily (USA Today, December 21, 2005).
a. In a sample of 10 workers, what is the probability that exactly three workers take public
transportation daily?
b. In a sample of 10 workers, what is the probability that at least three workers take public
transportation daily?

3. Auniversity found that 20% of its students withdraw without completing the introductory statistics
course. Assume that 20 students registered for the course.
a. Compute the probability that two or fewer will withdraw.
b. Compute the probability that exactly four will withdraw.
c. Compute the probability that more than three will withdraw.
d. Compute the expected number of withdrawals.

4. Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways.
a. Compute the probability of receiving three calls in a 5-minute interval of time.
b. Compute the probability of receiving exactly 10 calls in 15 minutes

.
5. More than 50 million guests stay at bed and breakfasts (B&Bs) each year. The website for the Bed
and Breakfast Inns of North America, which averages seven visitors per minute, enables many
B&Bs to attract guests (Time, September 2001).
a. Compute the probability of no website visitors in a one-minute period.
b. Compute the probability of two or more website visitors in a one-minute period.

6. In a survey conducted by the Gallup Organization, respondents were asked, “What is your favorite
sport to watch?” Football and basketball ranked number one and two in terms of preference
(Gallup website, January 3, 2004). Assume that in a group of 10 individuals, seven prefer football
and three prefer basketball. A random sample of three of these individuals is selected.
a. What is the probability that exactly two prefer football?
b. What is the probability that the majority (either two or three) prefer football?

Please answer these questions using SPSS. Thank you

Note: For all assignments, you must show the requested output from SPSS.

Example. Determine the descriptive statistics for three quantitative variables. Which variable has the highest mean? The most variability?

Answer: Output should include 3 boxes of descriptive statistics, one for each variable. There should also be one page that gives the answer to the other two questions.

The following sample data are used. We are interested in the descriptive statistics from these variables: sleep duration of undergrads, the number of NBA wins and the number of words in a résumé). Assume that they are all samples from populations. Use the instructions from the first two recitation worksheets to help you.

¹ This is a survey of the amount of sleep for one night for 40 undergraduates.

² These are the numbers of wins for the 30 National Basketball Association teams for the 2014-2015 NBA season.

³ A large university offers training on résumé construction. This lists the number of words in each of 23 résumés.

Note: Here are the 5% Trimmed Means for these variables. This will help you make sure you’ve input the data correctly: Amount of sleep: 419.94 minutes. Wins: 42.69. Words: 257.72.

1. Input the data the way you learned in recitation. You can input all of the data (3 data sets) at the same time, but you must run the analysis of each variable individually, otherwise SPSS won’t read it correctly, because the sample sizes are different. [Compare your descriptive statistics with the means and standard deviations on your graphs; this will help you make sure100405405 your numbers are correct.] In addition, I want you to add another variable next to the words variable which will divide the variable “Words” categorically. I want you to call this variable “wordscat.” (Note: this is the variable name. The variable labelcan be more descriptive, e.g., Words by Category). Create this variable using the Transform procedure you used in lab. Remember that you will have to give this variable Value Labels in the Variable View.

The scores should be divided into categories like this:

Low: 100-200              Moderate: 201-300           High: 301-400

2. For all the quantitative variables (wordscat is categorical) find descriptive statistics (using Explore). Print out “Descriptives” and Stemplot. Remember to do this for each variable individually. If you try to do them at the same time your answers will be incorrect.

3. For all the quantitative variables (wordscat is categorical) create and print out frequency distributions and histograms (using Frequencies). Remember to do this for each variable individually. If you try to do them at the same time your answers will be incorrect. Note: When doing the histogram, if you check the box “Show Normal Curve” this does not mean your variable is normal. It just shows what normal would look like and you can compare the shape of your variable to what is normal.

4. For the categorical variable (wordscat) create and print out a bar chart. Make sure the variable is categorical on your chart and remember to label the chart.

5. Create a variable (call it “sleeptwo”) that multiplies each student’s amount of sleep by 2 (do this by using Transform > Compute like in Lab 2, but instead of adding, we are multiplying by 2. Use this sign * for multiplication). Repeat steps 2 and 3 above for the new variable.

On your separate page please answer the following questions (worth 8 points):

6. Calculate the SS for each of the three main quantitative variables using the variance you obtained in SPSS (this should be on a sheet separate from the SPSS output). Note: Remember the relationship between variance and SS to help you do this.

7. Write 1-2 sentences (on your separate page) for each of the three main quantitative variables including the following: values of the mean, median, standard deviation and approximate shape of the distribution (look at the histogram). The shape may not be obvious; this is ok. Just do your best to describe what you think is going on with the data (e.g., mention whether you think there is skew, are there are any outliers, is it bimodal?).

8. Answer the following questions:
   1. What happened to the mean and standard deviation of sleeptwo compared to the original mean and standard deviation of that variable?
   2. What happened to the shape of the distribution of the sleep variable after multiplying by two (compared to the original variable)? What is the explanation for this?