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

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