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# Probability Using a Frequency Table & the Central Limit Theorem

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1. You roll a die, winning nothing if the number of spots is odd, \$3 for a 2 or a 4 and a \$18 for a 6.

a) Find the expected value and standard deviation of your prospective winnings
b) You play three times. Find the mean and standard deviation of your total winnings.
c) You play 60 times. What is the probability that you win at least \$310?

2. The score distribution shown in the table is for all students who took a yearly AP statistics exam.

Score % of students
5 13.3
4 22.9
3 25.5
2 17.1
1 21.2

a) Find the mean and standard deviation of the scores

b) Considering the mean scores of random samples of 40 AP statistics students. Describe the sampling model for these means (shape, center, and spread). Select the correct choices below and fill in any answer boxes in your choice.

I think it's this one : The sampling model is normal with m-y=____ and SD(y)= _____

3. Assume that the duration of human pregnancies can be described by a normal model with mean 264 days and standard deviation 19 days.
a) what percentage of pregnancies should last between 275 and 285 days? (round to one decimal place as needed)
b) suppose a certain obstetrician is currently providing prenatal care to 20 pregnant women. Let y represent the mean length of their pregnancies. According to the central limit theorem what is the mean and standard deviation of the normal model of the distribution of the sample mean y?
c) What is the probability that the mean duration of these patients pregnancies will be less than 258 days?

b) at least how many days should the longest 20% of all pregnancies last?

4. Carbon monoxide emissions for a certain kind of car vary with mean 3.778 g/mi and standard deviation 0.8 g/mi. A company has 60 of these cars in its fleet. Let y represent the mean CO level for the companies fleet.
a) what's the approximate model for the distribution of y? Explain
b) Estimate the probability that y is between 3.9 and 4.1 g/mi.
c) There is only a 10% chance that the fleets mean CO level is greater than what value.

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#### Solution Preview

1. You roll a die, winning nothing if the number of spots is odd, \$3 for a 2 or a 4 and a \$18 for a 6.
a) Find the expected value and standard deviation of your prospective winnings
Since there are 6 outcomes when you roll a die, each outcome has a probability of 1/6. So P(win \$3)=1/6+1/6=1/3, P(win \$18)=1/6 and P(win nothing)=1/6+1/6+1/6=1/2.
So expected value=3*1/3+18*1/6+0*1/2=4
So squared expected value=32*1/3+182*1/6+02*1/2=57
So standard deviation=sqrt(57-42)=sqrt(41)=6.4
b) You play three times. Find the mean and standard deviation of your total winnings.
Now you roll a die three times, mean=3*4=12.
And standard deviation=sqrt(3*41)=11.09
c) You play 60 times. What is the probability that you win at least \$310?
Now you roll a die 60 times, mean=60*4=240
And standard deviation=sqrt(60*41)=49.6
So P(win at least \$310)=P(Z≥(310-240)/49.6)=P(Z≥1.41)=0.0793 from standard normal table.

2. The score distribution shown in ...

#### Solution Summary

The solution gives detailed steps on solving 3 question on calculating probability using frequency table and central limit theorem.

\$2.19

## Normal Distributions, Central Limit Theorem and Age Distribution

Questions:

Work each question, using Excel where appropriate.

Remember, each gestation period is its own normal distribution. Thus, you will need to change the "mean" and "standard deviation" to reflect the question you are answering.

2. What percent of the babies born with each gestation period have a low birth weight (under 5.5 pounds)?
a. under 28 weeks
b. 32 to 35 weeks
c. 37 to 39 weeks
d. 42 weeks and over

3. Describe the weights of the top 10% of the babies born with each gestation period.
a. 37 to 39 weeks
b. 42 weeks and over

4. For each gestation period, what is the probability that a baby will weigh between 6 and 9 pounds at birth?
a. 32 to 35 weeks
b. 37 to 39 weeks
c. 42 weeks and over

5. A birth weight of less than 3.3 pounds is classified by the NCHS as a "very low birth weight." What is the probability that a baby has a very low birth weight for each gestation period?
a. under 28 weeks
b. 32 to 35 weeks
c. 37 to 39 weeks
Questions:

Remember, each gestation period is its own normal distribution. Thus, you will need to change the "mean" and "standard deviation" to reflect the question you are answering.

1. Enter the age distribution of the United States into a technology tool. Use the the tool to find the mean age in the United States.

2. Enter the set of sample means into a technology tool. Find the mean of the set of sample means. How does it compare with the mean age in the United States? Does this agree with the results by central Limit theorem?

3. Are the ages of people in the United States normally distributed? Explain your reasoning.

4. Sketch a relative frequency histogram for hte 36 sampl means. Use 9 classes. Is the histogram approximately bell shaped and symmetic? Does this agree with the results predicted by the Central Limite Theorem?

5. Use technology tool to find the standard deviation of the set of 36 samoke means. How does it compare with the standard deviation of the ages? Does this agree with the result predicted by the Central Limit Theorem?
Questions:

1. Using Excel, find the z-score that corresponds to the following Confidence Levels:

a. 80%
b. 85%
c. 92%
d. 97%

2. Using Excel, find the t-score that corresponds to the following Confidence Levels and Sample Sizes:

a. 95% with n = 25
b. 96% with n = 15
c. 97% with n = 21
d. 91% with n = 10

3. Suppose we wish to estimate the population mean using a confidence interval. When is it appropriate to use a z-score? When is it appropriate to use a t-score?

Bob loves making candy, especially varieties of caramel, including plain, chocolate dipped caramels and chocolate dipped caramels with pecans. Bob has received lots of compliments from his friends and neighbors, and several have encouraged him to start his own candy making business.

After several days of research, Bob finds that the national average amount of money spent annually per person on this type of specialty candy is \$75. Bob believes that the citizens in his area spend more than that per year. Knowing whether or not this is true could help Bob make a wise decision regarding his future business plans.

Bob wants to use statistics to support his claim, and to help him obtain a small business loan. Bob also wants to find an estimate of the true amount of money local citizens do spend on this type of specialty candy.

Bob randomly selects several people from his local phone book and asks the person that answers how much money they typically spend per year on candy like he will make. He obtains the following results (in dollars): 75, 74, 80, 68, 79, 85, 77, 82, 79, 67, 90, 72, 76, 75, 69, 85, 78, 79, 82, 66, 75, 85, 90, 76, 85, 67, 89, 82, 69, 79, 82, 80, 84, 79, 78, 81, 77, 84, 80, 76.

Based upon these results, Bob is hoping his area has a good customer base for his new business. Bob also hopes the bank is impressed with his use of statistics and will grant him the loan he needs to start it!

Questions:
1. Find the sample mean and sample standard deviation of the amount citizens spend per year.

2. When finding a confidence interval for the true mean spent of ALL citizens, should we use a z-score or
a t-score? Why?

3. Find the z/t-values (as appropriate) for a 95% confidence interval and a 92% confidence interval.

4. Find a 95% and a 92% confidence interval for the true mean amount that citizens spend per year.

5. What do you think the lowest possible mean amount spent per year is? Why?

6. Do you think Bob has a good customer base for his new business? Explain.

Please see the 4 tabs on this spreadsheet. Specifically, Birth Weights,Age Dist, Conf Intervals & Candy Business

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