Download Sampling Distribution with CLT Worksheet

Sampling Distribution with CLT Worksheet
On AP Stat Formula Chart (for exam)
If X has a binomial distribution with parameters n and p, then:
 x  np
 x  np (1  p )
 pˆ  p
p (1  p )
n
 pˆ 
If x is the mean of a random sample of size n from an infinite
Population with mean  and standard deviation , then:
x  
x 

n
Exercises
1. What is the difference between a proportion and a mean?
2. A sample is chosen randomly from a population that can be described as normally distributed.
a) What is the sampling distribution shape, center, and spread compared to the population?
b) If we choose a larger sample, what is the effect on this sampling distribution? What stays the
same, what changes and how?
3. A sample is chosen randomly from a population that is strongly skewed right, …
a) Describe the sampling distribution for sample mean if the sample size was small.
b) As we make the sample size larger, what happens to the expected sampling distribution of
sample mean’s shape, center, and spread compared to the population’s.
4. M&M’s claims 20% of the candies it produces is red. Suppose that the candies are packaged at
random in small bags containing about 50 M&M’s. A class of elementary school students learning
about percents opens several bags, counts the red candies, and divides the count by the sample size of
50 candies, and calculates the proportion that are red.
a) If we plot a histogram showing the proportions of red candies in the various bags, what shape
would you expect it to have?
b) Can that histogram be approximated by a normally distributed sampling distribution? Explain
(verify). Why is this important? What does it tell us?
c) Where should the center of the histogram be?
d) What should the standard deviation of the sampling proportion distribution be?
5. State police believe that 70% of the drivers traveling on I-10 exceed the speed limit. They plan to set
up a radar trap and check the speeds of 80 cars.
a) What is the expected proportion of speeding cars?
b) What is the standard deviation of the proportion of speeding cars?
c) Verify if the appropriate conditions are met. Why is this important? What does it tell us?
d) If conditions have been met (which they should have), sketch and label a normal distribution
with mean and standard deviation from parts a and b.
e) What percent (probability) of the time will state police find more than 60 of the drivers
speeding (Hint: 60 of 80 is equivalent to 75%)? Approximate answer using 68-95-99.7% Rule
or use normalcdf command.
6. State police believe that 70% of the drivers traveling on I-10 exceed the speed limit. They plan to set
up a radar trap and check the speeds of 80 cars.
a) What is the expected number of speeding cars?
b) What is the standard deviation of the number of speeding cars?
c) Verify if the appropriate conditions are met. Why is this important? What does it tell us?
d) If conditions have been met (which they should have), sketch and label a normal distribution
with mean and standard deviation from parts a and b.
e) What percent (probability) of the time will state police find more than 60 of the drivers
speeding? Approximate answer using 68-95-99.7% Rule or use normalcdf command.
7. A college’s data about incoming freshmen indicates that the mean of their high school GPAs was 3.4
with a standard deviation of 0.35; the distribution was roughly mound-shaped and only slightly
skewed. The students are randomly assigned to freshman writing seminars in groups of 25.
a) What might the mean GPA of one group of these seminar groups be?
b) What is the approximate standard deviation of GPA of one group of these seminar groups?
c) Describe the appropriate shape of the sampling distribution of GPA of one group.
d) Is a sample size of 25 sufficiently large enough? Explain.
e) Sketch and label the sampling distribution of GPAs of one group.
f) What is the likelihood that one group would have a GPA mean less than 3.26?
8. The duration of human pregnancies can be described by a Normal model with mean 266 days and
standard deviation 16 days.
a) What percent of pregnancies should last between 270 and 280 days? (Use normalcdf).
b) At least how many days should the longest 25% of all pregnancies last? (Use invNorm).
c) Suppose an obstetrician is currently providing prenatal care to 60 pregnant women. Let x
represent the mean length of their pregnancies. According to the Central Limit Theorem, what
is the sampling distribution of this sample mean, x ? Specify the shape of model, center, and
spread (standard deviation).
d) What is the probability that the mean duration of these patient’s pregnancies will be less than
260 days?
e) If I were to now tell you that the duration of human pregnancies may actually be slightly
skewed to the left, would that change your answers to parts a and b?
f) If I were to now tell you that the duration of human pregnancies may actually be slightly
skewed to the left, would that change your answers to parts c and d?
g) Explain why the difference in answers to parts e and f.
9. Some business analysts estimate that the length of time people work at a job has a mean of 6.2 years
and a standard deviation of 4.5 years.
a) Explain why you suspect this distribution may be skewed to the right.
b) Explain why you could estimate the probability that 100 people selected at random had
worked for their employers an average of 10 years or more, BUT you could not estimate the
probability that an individual had done so. (Do not calculate probability—simply answer the
question).