6/17/24, 2:28
PM
Week 6 Homework
Terms in this set (77)
What is the sampling distribution of a A probability distribution for all possible values of the statistic computed from a sample of size n.
statistic?
What is the sampling distribution of the sample All possible values of the random variable computed from a sample size of n from a
mean overbar x? population with the mean µ and standard deviation σ.
1. Obtain a simple random sample of size n.
What are the three steps for determining the 2. Compute the sample mean.
sample distribution of the sample mean?
3. Assuming that we are sampling from a finite population, repeat steps 1 and 2 until all distinct
simple random samples of size n have been obtained.
What is the shape of the distribution of the It becomes approximately normal as the sample size, n, increases, regardless of the shape of the
sample mean as the sample size increases? underlying population.
n>30 for the sample mean (overbar x) to be normal if the parent population is not known to be
What is the Central Limit Theorem? normal.
Center: As the sample size n increases, what does It is equal to the mean of the underlying population. (The center is not affected.)
the mean of the distribution of the sample mean,
overbar x, equal?
Spread (σ sub-overbar x): As the sample size n As n increases, the standard deviation will decrease.
increases, what happens to the standard deviation
of the distribution of the sample mean?
Mean: µ sub-overbar x = µ
List the formulas for the mean, standard deviation,
and checking for independence of the sampling Standard deviation: σ sub-overbar x = σ/√n
distribution of overbar x.
Independence: n≤0.05N
Week 6 Homework
1/16
,6/17/24, 2:28
PM
Suppose a simple random sample of size n is The distribution becomes approximately normal.
obtained from a population whose distribution is
skewed right. As the sample size n increases, what According to the Central Limit Theorem, if the mean values for increasing sample sizes are
happens to the shape of the distribution of the obtained, the distribution of sample means will be normally distributed, even if the individual
sample mean?
samples do not have normal distributions. Typically, sample sizes of 30 or greater are
recommended.
A simple random sample of size n=31 is obtained No. The central limit theorem states that regardless of the shape of the underlying
from a population that is skewed left with μ=70 population, the sampling distribution of overbar x becomes approximately normal as
and σ=99. the sample size, n, increases.
Does the population need to be normally
distributed for the sampling distribution of
overbar x to be approximately normally
distributed? Why? What is the sampling
distribution of overbar x?
As the sample size n increases, what The standard error of the mean decreases.
happens to the standard error of the
mean?
Complete parts (a) through (d) for the sampling (a) 400
distribution of the sample mean shown in the
accompanying graph.
(b) 10
(a) What is the value of μ v overbar x?
(c) The shape of the population is approximately normal.
(b) What is the value of σ v overbar x?
(d) (√16) * 10 = 40
(c) If the sample size is n=16, what is likely
true about the shape of the population?
(d) If the sample size is n=16, what is the standard
deviation of the population from which the sample
was drawn?
The standard deviation of the population from
which the sample was drawn is 4040.
Week 6 Homework
2/16
, 6/17/24, 2:28 Week 6 Homework Flashcards |
PM Quizlet
Suppose a simple random sample of size n=37 is (a) Since the sample size is large enough, the population distribution does not need to be
obtained from a population with μ=67 and σ=17. normal.
(a) What must be true regarding the distribution (b) Approximately normal, with µ v overbar x=67 and σ v
of the population in order to use the normal model overbar x=17/(√37)
to compute probabilities regarding the sample
mean?
(c) normalcdf (-9999999, 71.1, 67, (17/(√37)) = 0.9288
(d) normalcdf (69.1, 9999999, 67, (17/√(37)) = 0.2262
(b) Assuming the normal model can be used,
describe the sampling distribution overbar x.
(c) Assuming the normal model can be used,
determine P(overbar x<71.1).
(d) Assuming the normal model can be used,
determine P(overbar x≥69.1).
Suppose the lengths of the pregnancies of a (a) normalcdf (-9999999, 203, 206, 10) = 0.3821
certain animal are approximately normally
distributed with mean μ=206 days If 100 pregnant individuals were selected independently from this population, we would expect 38
and standard deviation σ=10 days. pregnancies to last less than 203 days.
(a) What is the probability that a randomly (b) n = 23, σ = σ/√n = 10/√23 = 2.0851
selected pregnancy lasts less than 203 days?
Interpret this probability. The sampling distribution of overbar x is normal with μx=206 and σx=2.0851 (c) normalcdf (-
(b) Suppose a random sample of 23 pregnancies is 9999999, 203, 206, (10/(√23)) = 0.0751
obtained. Describe the sampling distribution of the
sample mean length of pregnancies. If 100 independent random samples of size n=23
pregnancies were obtained from this population, we would expect 8 sample(s) to have a sample
(c) What is the probability that a random sample mean of
of 23 203 days or less.
pregnancies has a mean gestation period of 203
days or less? Interpret this probability.
(d) normalcdf (-9999999, 203, 206, (10/(√50)) = 0.0169
(d) What is the probability that a random sample
of 50 If 100 independent random samples of size n=50
pregnancies has a mean gestation period of 203 pregnancies were obtained from this population, we would expect 2 sample(s) to have a sample
days or less? Interpret this probability. mean of 203 days or less.
(e) What might you conclude if a random (e) This result would be unusual, so the sample likely came from a population whose mean
sample of 50 gestation period is less than 206 days.
pregnancies resulted in a mean gestation period of
203 days or less? (f) Within 9 days of the mean = 197 to 215 days
(f) What is the probability a random sample normalcdf (197, 215, 206, (10/√19)) = 0.9999
of size 19 will have a mean gestation period
within 9 days of the
Week
mean 6 Homework
?
3/16
PM
Week 6 Homework
Terms in this set (77)
What is the sampling distribution of a A probability distribution for all possible values of the statistic computed from a sample of size n.
statistic?
What is the sampling distribution of the sample All possible values of the random variable computed from a sample size of n from a
mean overbar x? population with the mean µ and standard deviation σ.
1. Obtain a simple random sample of size n.
What are the three steps for determining the 2. Compute the sample mean.
sample distribution of the sample mean?
3. Assuming that we are sampling from a finite population, repeat steps 1 and 2 until all distinct
simple random samples of size n have been obtained.
What is the shape of the distribution of the It becomes approximately normal as the sample size, n, increases, regardless of the shape of the
sample mean as the sample size increases? underlying population.
n>30 for the sample mean (overbar x) to be normal if the parent population is not known to be
What is the Central Limit Theorem? normal.
Center: As the sample size n increases, what does It is equal to the mean of the underlying population. (The center is not affected.)
the mean of the distribution of the sample mean,
overbar x, equal?
Spread (σ sub-overbar x): As the sample size n As n increases, the standard deviation will decrease.
increases, what happens to the standard deviation
of the distribution of the sample mean?
Mean: µ sub-overbar x = µ
List the formulas for the mean, standard deviation,
and checking for independence of the sampling Standard deviation: σ sub-overbar x = σ/√n
distribution of overbar x.
Independence: n≤0.05N
Week 6 Homework
1/16
,6/17/24, 2:28
PM
Suppose a simple random sample of size n is The distribution becomes approximately normal.
obtained from a population whose distribution is
skewed right. As the sample size n increases, what According to the Central Limit Theorem, if the mean values for increasing sample sizes are
happens to the shape of the distribution of the obtained, the distribution of sample means will be normally distributed, even if the individual
sample mean?
samples do not have normal distributions. Typically, sample sizes of 30 or greater are
recommended.
A simple random sample of size n=31 is obtained No. The central limit theorem states that regardless of the shape of the underlying
from a population that is skewed left with μ=70 population, the sampling distribution of overbar x becomes approximately normal as
and σ=99. the sample size, n, increases.
Does the population need to be normally
distributed for the sampling distribution of
overbar x to be approximately normally
distributed? Why? What is the sampling
distribution of overbar x?
As the sample size n increases, what The standard error of the mean decreases.
happens to the standard error of the
mean?
Complete parts (a) through (d) for the sampling (a) 400
distribution of the sample mean shown in the
accompanying graph.
(b) 10
(a) What is the value of μ v overbar x?
(c) The shape of the population is approximately normal.
(b) What is the value of σ v overbar x?
(d) (√16) * 10 = 40
(c) If the sample size is n=16, what is likely
true about the shape of the population?
(d) If the sample size is n=16, what is the standard
deviation of the population from which the sample
was drawn?
The standard deviation of the population from
which the sample was drawn is 4040.
Week 6 Homework
2/16
, 6/17/24, 2:28 Week 6 Homework Flashcards |
PM Quizlet
Suppose a simple random sample of size n=37 is (a) Since the sample size is large enough, the population distribution does not need to be
obtained from a population with μ=67 and σ=17. normal.
(a) What must be true regarding the distribution (b) Approximately normal, with µ v overbar x=67 and σ v
of the population in order to use the normal model overbar x=17/(√37)
to compute probabilities regarding the sample
mean?
(c) normalcdf (-9999999, 71.1, 67, (17/(√37)) = 0.9288
(d) normalcdf (69.1, 9999999, 67, (17/√(37)) = 0.2262
(b) Assuming the normal model can be used,
describe the sampling distribution overbar x.
(c) Assuming the normal model can be used,
determine P(overbar x<71.1).
(d) Assuming the normal model can be used,
determine P(overbar x≥69.1).
Suppose the lengths of the pregnancies of a (a) normalcdf (-9999999, 203, 206, 10) = 0.3821
certain animal are approximately normally
distributed with mean μ=206 days If 100 pregnant individuals were selected independently from this population, we would expect 38
and standard deviation σ=10 days. pregnancies to last less than 203 days.
(a) What is the probability that a randomly (b) n = 23, σ = σ/√n = 10/√23 = 2.0851
selected pregnancy lasts less than 203 days?
Interpret this probability. The sampling distribution of overbar x is normal with μx=206 and σx=2.0851 (c) normalcdf (-
(b) Suppose a random sample of 23 pregnancies is 9999999, 203, 206, (10/(√23)) = 0.0751
obtained. Describe the sampling distribution of the
sample mean length of pregnancies. If 100 independent random samples of size n=23
pregnancies were obtained from this population, we would expect 8 sample(s) to have a sample
(c) What is the probability that a random sample mean of
of 23 203 days or less.
pregnancies has a mean gestation period of 203
days or less? Interpret this probability.
(d) normalcdf (-9999999, 203, 206, (10/(√50)) = 0.0169
(d) What is the probability that a random sample
of 50 If 100 independent random samples of size n=50
pregnancies has a mean gestation period of 203 pregnancies were obtained from this population, we would expect 2 sample(s) to have a sample
days or less? Interpret this probability. mean of 203 days or less.
(e) What might you conclude if a random (e) This result would be unusual, so the sample likely came from a population whose mean
sample of 50 gestation period is less than 206 days.
pregnancies resulted in a mean gestation period of
203 days or less? (f) Within 9 days of the mean = 197 to 215 days
(f) What is the probability a random sample normalcdf (197, 215, 206, (10/√19)) = 0.9999
of size 19 will have a mean gestation period
within 9 days of the
Week
mean 6 Homework
?
3/16
