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Probability: Normal Distribution and The Central Limit Theorem

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1. The weight of potato chips in a small size bag is stated to be 5 ounces. The amount that the packaging machine puts in these bags is believed to have a normal model with a mean of 5.1 ounces and a standard deviation of 0.08 ounces.
a) What fraction of all bags sold are underweight?
b) Some of the chips are sold in "bargain packs" of 3 bags. What's the probability that none of the 3 is underweight?
c) What's the probability that the mean weight of the 3 bags is below the stated amount?
d) What's the probability that the mean weight of a 24-bag case of potato chips is below 5oz?

2. Suppose that the IQ's of university A's students can be described by a normal model with mean 140 and standard deviation 7 points. Also suppose that IQ of students from university B can be described by a normal model with mean 110 and standard deviation 9.
a) The probability is __________.
b) We select a student at random from each school. Find the probability that the university A student's IQ is at least 5 points higher than the university B students IQ.
c) Select 3 university B students at random. Find the probability that this groups average IQ is at least 130 points.

3. The data in the table represents the ages of the winners of an award for the past five years. Use the data to answer questions.
37 42
41 45
46
a) Find the population mean age of the five winners
b) For a sample size of three construct a table of all possible samples and their sample means.
c) Draw a dot plot for the sampling of the sample mean for the sample size of three.
d) For a random sample size 3. What is the chance that the sample mean will equal the population mean.
e) For a random sample size 3, obtain the probability that the sampling error made in estimating the population mean by the sample mean will be 1 year or less, that is determine the probability that x be within 1 year of the mean.

4. Complete parts (a) through (e) for the population data below 3, 4, 5
a) Find the mean, m, of the variable
b) For each of the possible sample sizes construct a table with all possible samples and their sample means, and draw a dotplot for sampling distribution of the sample mean. The sample mean x is found by summing the observations in the sample and dividing by the sample size.
c) Use the dotplots from part (b) to create one plot with the sampling distributions for each sample size and interpret your results. The correct plot is shown below.
d) For each of the possible sample sizes find the probability that the sample mean will equal the population mean.
e) For each of the possible sample sizes find the probability that the sampling error made in estimating the population mean by the sample mean will be 0.5 or less (in magnitude) that is that the absolute value of the difference between the sample mean and the population mean is almost 0.5.

5. The data in the table represent the ages of the winners of an award for the past five years. Use the data to answer questions (a) through (c).
a) Determine the population mean age, M, of the five numbers.
b) Consider a sample size of 2 without replacement. Find the mean of the variable x

6. A variable of a population has a mean of m=73 and a standard deviation of x=7.
a) Identify the sampling distribution of the sample mean for samples of size 49.
b) In answering part (a) what assumptions did you make about the distribution of the variable?
c) Can you answer part (a) if the sample size is 25 instead of 49? Why or why not?
d) What is the shape of the sampling distribution? Uniform, skewed, normal, bimodal?
e) What is the mean of the sampling distribution?
f) What is the standard deviation of the sampling distribution?

7. According to one study brain weights of men are normally distributed with a mean of 1.60 kg and standard deviation of 0.12kg. Use the data to answer questions (a) through (e).
a) Determine the sampling distribution of the sample mean for samples of size 3. The sample mean is m-=
b) Determine the sampling distribution of the sample mean for samples of size 12.
c) Construct a graph of the normal population distribution and the two sampling distributions for brain weights.
d) Determine the percentage of all samples of three men that have mean brain weights within 0.1kg of the population mean brain weights of 1.70kg.
e) Determine the percentage of all samples of twelve men that have mean brain weights within 0.1kg the population mean brain weights of 1.70kg.

8. The mean annual salary of classroom teachers is approximately normal with a mean of $45,800. Assume a standard deviation of $8800. Do the following for the variable "annual salary" of classroom teachers.
a) Determine the sampling distribution of the sample mean for samples of size 64.
b) The sample mean is?
c) Determine the sampling of the sample mean for sample sizes 256
d) What is the probability that the sampling made in estimating the population mean salary of all classroom teachers by the mean salary of a sample of 64 classroom teachers will be at most $1000?
e) What is the probability that the sampling error made estimating the population mean salary of all classroom teachers by the mean salary of a sample of 256 classroom teachers will be at most $1000?

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Solution Summary

The solution gives detailed steps on solving 8 questions by finding the probability under normal distribution by applying central limit theorem.

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Please check the attachment for the solutions. Questions 3 to 5 are missing data.

1. The weight of potato chips in a small size bag is stated to be 5 ounces. The amount that the packaging machine puts in these bags is believed to have a normal model with a mean of 5.1 ounces and a standard deviation of 0.08 ounces.
a) What fraction of all bags sold are underweight?
P(X<5)=P(Z<(5-5.1)/0.08)=P(Z<-1.25)=0.1056 from normal table
b) Some of the chips are sold in "bargain packs" of 3 bags. What's the probability that none of the 3 is underweight?
P(none of 3 is underweight)=(1-0.1056)3=0.7155
c) What's the probability that the mean weight of the 3 bags is below the stated amount?
P(xbar<5)=P(Z<(5-5.1)/(0.08/sqrt(3))=P(Z<-2.17)=0.0150 from normal table
d) What's the probability that the mean weight of a 24-bag case of potato chips is below 5oz?
P(xbar<5)=P(Z<(5-5.1)/(0.08/sqrt(24))=P(Z<-6.12)=0.0000 from normal table
2. Suppose that the IQ's of university A's students can be described by a normal model with mean 140 and standard deviation 7 points. Also suppose that IQ of students from university B can be described by a normal model with mean 110 and standard deviation 9.
B) We select a student at random from each school. Find the probability that the university A student's IQ is at least 5 points higher than the university B students IQ.
First, the mean difference between two universities are normal with mean 140-110=30 and variance 72+92=140
So P(IQ for A is at least 5 points higher than B)=P(Z>(5-30)/sqrt(140))=P(Z>-2.11)=0.9826 from normal table.
c) select 3 university B students at random. Find the probability that this groups average IQ is at least 130 points.
P(X>130)=P(Z>(130-110)/(9/sqrt(3)))=P(Z>3.85)=0.0001 from normal table
3. The data in the table represents the ages of the winners of an award for the past five years. Use the data to answer questions.
37 42
41 ...

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