Transcript Increase the margin of error.

Introduction to Inferece
BPS chapter 14
© 2010 W.H. Freeman and Company
Statistical Inference
What is statistical inference on ?
a)
b)
c)
d)
Drawing conclusions about a population mean based on information
contained in a sample.
Drawing conclusions about a sample mean based on information
contained in a population.
Drawing conclusions about a sample mean based on the
measurements in that sample.
Selecting a set of data from a large population.
Statistical Inference (answer)
What is statistical inference on ?
a)
b)
c)
d)
Drawing conclusions about a population mean based on
information contained in a sample.
Drawing conclusions about a sample mean based on information
contained in a population.
Drawing conclusions about a sample mean based on the
measurements in that sample.
Selecting a set of data from a large population.
Inference
The conditions for doing inference on  using the standard normal
distribution do NOT include:
a)
b)
c)
d)
A simple random sample of size n.
A normal population or sample size large enough to apply the
Central Limit Theorem.
A known value of .
A known value of .
Inference (answer)
The conditions for doing inference on  using the standard normal
distribution do NOT include:
a)
b)
c)
d)
A simple random sample of size n.
A normal population or sample size large enough to apply the
Central Limit Theorem.
A known value of .
A known value of .
Inference
Why do we need a normal population or large sample size to do
inference on ?
a)
b)
c)
d)
So that the sampling distribution of x is normal or approximately
normal.
So that the distribution of the sample data is normal or
approximately normal.
So that x equals .
So that  is known.
Inference (answer)
Why do we need a normal population or large sample size to do
inference on ?
a)
b)
c)
d)
So that the sampling distribution of x is normal or
approximately normal.
So that the distribution of the sample data is normal or
approximately normal.
So that x equals .
So that  is known.
Inference
True or False: The condition of known  is often met even when  is
unknown.
a)
b)
True
False
Inference (answer)
True or False: The condition of known  is often met even when  is
unknown.
a)
b)
True
False
Confidence Intervals
The purpose of a confidence interval for  is
a)
b)
c)
d)
To give a range of reasonable values for the level of confidence.
To give a range of reasonable values for the sample mean.
To give a range of reasonable values for the population mean.
To give a range of reasonable values for the difference between the
sample mean and the population mean.
Confidence Intervals (answer)
The purpose of a confidence interval for  is
a)
b)
c)
d)
To give a range of reasonable values for the level of confidence.
To give a range of reasonable values for the sample mean.
To give a range of reasonable values for the population mean.
To give a range of reasonable values for the difference between the
sample mean and the population mean.
Confidence intervals
The confidence interval formula for  does NOT include
a)
b)
c)
d)
e)
f)
The sample mean.
The population standard deviation.
The z* value for specified level of confidence.
The margin of error.
The sample size.
The population size.
Confidence intervals (answer)
The confidence interval formula for  does NOT include
a)
b)
c)
d)
e)
f)
The sample mean.
The population standard deviation.
The z* value for specified level of confidence.
The margin of error.
The sample size.
The population size.
Confidence intervals
What do we hope to capture within a confidence interval?
a)
b)
c)
d)
e)
f)
The unknown confidence level.
The unknown parameter.
The unknown statistic.
The parameter estimate.
The margin of error.
The sample size.
Confidence intervals (answer)
What do we hope to capture within a confidence interval?
a)
b)
c)
d)
e)
f)
The unknown confidence level.
The unknown parameter.
The unknown statistic.
The parameter estimate.
The margin of error.
The sample size.
Confidence intervals
What are the three components of a confidence interval?
a)
b)
c)
Estimate of confidence level, sample size, and margin of error.
Mean of sample statistic, confidence level, and margin of error.
Estimate of population parameter, confidence level, and margin of
error.
Confidence intervals (answer)
What are the three components of a confidence interval?
a)
b)
c)
Estimate of confidence level, sample size, and margin of error.
Mean of sample statistic, confidence level, and margin of error.
Estimate of population parameter, confidence level, and margin
of error.
Confidence intervals
Consider the following statement.
“The average time a local company takes to process new insurance
claims is 9 to 11 days.”
Which of the following statements about this confidence interval
interpretation is valid?
a)
b)
c)
d)
This interpretation is incorrect because there is no statement of the
true parameter in words.
This interpretation is incorrect because the interval is not reported.
This interpretation is incorrect because the confidence level is not
reported.
This is a correct reporting of the confidence interval.
Confidence intervals (answer)
Consider the following statement.
“The average time a local company takes to process new insurance
claims is 9 to 11 days.”
Which of the following statements about this confidence interval
interpretation is valid?
a)
b)
c)
d)
This interpretation is incorrect because there is no statement of the
true parameter in words.
This interpretation is incorrect because the interval is not reported.
This interpretation is incorrect because the confidence level is
not reported.
This is a correct reporting of the confidence interval.
Confidence intervals
A very large school district in Connecticut wants to estimate the
average SAT score of this year’s graduating class. The district
takes a simple random sample of 100 seniors and calculates the
95% confidence interval for the graduating students’ average SAT
score at 505 to 520 points.
For the sample of 100 graduating seniors, 95% of their SAT scores
were between 505 and 520 points.
a)
b)
Correct interpretation of interval.
Incorrect interpretation of interval.
Confidence intervals (answer)
A very large school district in Connecticut wants to estimate the
average SAT score of this year’s graduating class. The district
takes a simple random sample of 100 seniors and calculates the
95% confidence interval for the graduating students’ average SAT
score at 505 to 520 points.
For the sample of 100 graduating seniors, 95% of their SAT scores
were between 505 and 520 points.
a)
b)
Correct interpretation of interval.
Incorrect interpretation of interval.
Confidence intervals
The probability that a 90% confidence interval for a population mean
captures  is
a)
b)
c)
d)
0.
0.90.
1.
Either 0 or 1—we do not know which.
Confidence intervals (answer)
The probability that a 90% confidence interval for a population mean
captures  is
a)
b)
c)
d)
0.
0.90.
1.
Either 0 or 1—we do not know which.
Confidence intervals
What is the confidence level in a confidence interval for ?
a)
b)
c)
The percentage of confidence intervals produced by the procedure
that contain .
The probability that a specific confidence interval contains .
The percentage of confidence interval procedures that will create an
interval that contains .
Confidence intervals (answer)
What is the confidence level in a confidence interval for ?
a)
b)
c)
The percentage of confidence intervals produced by the
procedure that contain .
The probability that a specific confidence interval contains .
The percentage of confidence interval procedures that will create an
interval that contains .
Margin of error
Increasing the confidence level will
a)
b)
Increase the margin of error.
Decrease the margin of error.
Margin of error (answer)
Increasing the confidence level will
a)
b)
Increase the margin of error.
Decrease the margin of error.
Margin of error
Increasing the sample size will
a)
b)
Increase the margin of error.
Decrease the margin of error.
Margin of error (answer)
Increasing the sample size will
a)
b)
Increase the margin of error.
Decrease the margin of error.
Margin of error
Increasing the standard deviation will
a)
b)
Increase the margin of error.
Decrease the margin of error.
Margin of error (answer)
Increasing the standard deviation will
a)
b)
Increase the margin of error.
Decrease the margin of error.
Margin of error
Which of the following components of the margin of error in a
confidence interval for  does a researcher NOT have the chance to
select?
a)
b)
c)
Confidence level.
Sample size.
Population standard deviation.
Margin of error (answer)
Which of the following components of the margin of error in a
confidence interval for  does a researcher NOT have the chance to
select?
a)
b)
c)
Confidence level.
Sample size.
Population standard deviation.
Statistical significance
A consumer advocate is interested in evaluating the claim that a new
granola cereal contains “4 ounces of cashews in every bag.” The
advocate recognizes that the amount of cashews will vary slightly
from bag to bag, but she suspects that the mean amount of
cashews per bag is less than 4 ounces. To check the claim, the
advocate purchases a random sample of 40 bags of cereal and
calculates a sample mean of 3.68 ounces of cashews.
What is the size of the observed effect?
a)
b)
c)
d)
3.68 oz.
-3.68 oz.
4 – 3.68 = 0.32 oz.
3.68 – 4 = - 0.32 oz.
Statistical significance (answer)
A consumer advocate is interested in evaluating the claim that a new
granola cereal contains “4 ounces of cashews in every bag.” The
advocate recognizes that the amount of cashews will vary slightly
from bag to bag, but she suspects that the mean amount of
cashews per bag is less than 4 ounces. To check the claim, the
advocate purchases a random sample of 40 bags of cereal and
calculates a sample mean of 3.68 ounces of cashews.
What is the size of the observed effect?
a)
b)
c)
d)
3.68 oz.
-3.68 oz.
4 – 3.68 = 0.32 oz.
3.68 – 4 = - 0.32 oz.
Statistical significance
A consumer advocate is interested in evaluating the claim that a new
granola cereal contains “4 ounces of cashews in every bag.” The
advocate recognizes that the amount of cashews will vary slightly
from bag to bag, but she suspects that the mean amount of
cashews per bag is less than 4 ounces. To check the claim, the
advocate purchases a random sample of 40 bags of cereal and
calculates a sample mean of 3.68 ounces of cashews.
Suppose the consumer advocate computes the probability described in
the previous question to be 0.0048. Her result is
a)
b)
Statistically significant.
Not statistically significant.
Statistical significance (answer)
A consumer advocate is interested in evaluating the claim that a new
granola cereal contains “4 ounces of cashews in every bag.” The
advocate recognizes that the amount of cashews will vary slightly
from bag to bag, but she suspects that the mean amount of
cashews per bag is less than 4 ounces. To check the claim, the
advocate purchases a random sample of 40 bags of cereal and
calculates a sample mean of 3.68 ounces of cashews.
Suppose the consumer advocate computes the probability described in
the previous question to be 0.0048. Her result is
a)
b)
Statistically significant.
Not statistically significant.
Stating hypotheses
A consumer advocate is interested in evaluating the claim that a new
granola cereal contains “4 ounces of cashews in every bag.” The
advocate recognizes that the amount of cashews will vary slightly
from bag to bag, but she suspects that the mean amount of
cashews per bag is less than 4 ounces. To check the claim, the
advocate purchases a random sample of 40 bags of cereal and
calculates a sample mean of 3.68 ounces of cashews.
What alternative hypothesis does she want to test?
a)
b)
c)
d)
e)
f)
Stating hypotheses (answer)
A consumer advocate is interested in evaluating the claim that a new
granola cereal contains “4 ounces of cashews in every bag.” The
advocate recognizes that the amount of cashews will vary slightly
from bag to bag, but she suspects that the mean amount of
cashews per bag is less than 4 ounces. To check the claim, the
advocate purchases a random sample of 40 bags of cereal and
calculates a sample mean of 3.68 ounces of cashews.
What alternative hypothesis does she want to test?
a)
b)
c)
d)
e)
f)
Statistical significance
We reject the null hypothesis whenever
a)
b)
c)
d)
P-value > .
P-value  .
P-value  .
P-value  .
Statistical significance (answer)
We reject the null hypothesis whenever
a)
b)
c)
d)
P-value > .
P-value  .
P-value  .
P-value  .
Statistical significance
The significance level is denoted by
c)



d)
P-value
a)
b)
Statistical significance (answer)
The significance level is denoted by
c)



d)
P-value
a)
b)
Statistical significance
Which of the following is a conservative choice for significance level?
a)
b)
c)
d)
e)
f)
0
0.01
0.25
0.50
0.75
1
Statistical significance (answer)
Which of the following is a conservative choice for significance level?
a)
b)
c)
d)
e)
f)
0
0.01
0.25
0.50
0.75
1
Calculating P-values
To calculate the P-value for a significance test, we need to use
information about the
a)
b)
c)
Sample distribution
Population distribution
Sampling distribution of
Calculating P-values (answer)
To calculate the P-value for a significance test, we need to use
information about the
a)
b)
c)
Sample distribution
Population distribution
Sampling distribution of
Conclusions
Suppose the P-value for a hypothesis test is 0.304. Using  = 0.05,
what is the appropriate conclusion?
a)
b)
c)
d)
Reject the null hypothesis.
Reject the alternative hypothesis.
Do not reject the null hypothesis.
Do not reject the alternative hypothesis.
Conclusions (answer)
Suppose the P-value for a hypothesis test is 0.304. Using  = 0.05,
what is the appropriate conclusion?
a)
b)
c)
d)
Reject the null hypothesis.
Reject the alternative hypothesis.
Do not reject the null hypothesis.
Do not reject the alternative hypothesis.
Conclusions
Suppose the P-value for a hypothesis test is 0.0304. Using  = 0.05,
what is the appropriate conclusion?
a)
b)
c)
d)
Reject the null hypothesis.
Reject the alternative hypothesis.
Do not reject the null hypothesis.
Do not reject the alternative hypothesis.
Conclusions (answer)
Suppose the P-value for a hypothesis test is 0.0304. Using  = 0.05,
what is the appropriate conclusion?
a)
b)
c)
d)
Reject the null hypothesis.
Reject the alternative hypothesis.
Do not reject the null hypothesis.
Do not reject the alternative hypothesis.
P-value
True or False: The P-value should be calculated BEFORE choosing
the significance level for the test.
a)
b)
True
False
P-value (answer)
True or False: The P-value should be calculated BEFORE choosing
the significance level for the test.
a)
b)
True
False
Conclusions
Suppose a significance test is being conducted using a significance
level of 0.10. If a student calculates a P-value of 1.9, the student
a)
b)
c)
Should reject the null hypothesis.
Should fail to reject the null hypothesis.
Made a mistake in calculating the P-value.
Conclusions (answer)
Suppose a significance test is being conducted using a significance
level of 0.10. If a student calculates a P-value of 1.9, the student
a)
b)
c)
Should reject the null hypothesis.
Should fail to reject the null hypothesis.
Made a mistake in calculating the P-value.
Stating hypotheses
If we test H0:  = 40 vs. Ha:  < 40, this test is
a)
b)
c)
One-sided (left tail).
One-sided (right tail).
Two-sided.
Stating hypotheses (answer)
If we test H0:  = 40 vs. Ha:  < 40, this test is
a)
b)
c)
One-sided (left tail).
One-sided (right tail).
Two-sided.
Stating hypotheses
If we test H0:  = 40 vs. Ha:   40, this test is
a)
b)
c)
One-sided (left tail).
One-sided (right tail).
Two-sided.
Stating hypotheses (answer)
If we test H0:  = 40 vs. Ha:   40, this test is
a)
b)
c)
One-sided (left tail).
One-sided (right tail).
Two-sided.
Using CIs to test
A researcher is interested in estimating the mean yield (in bushels per
acre) of a variety of corn. From her sample, she calculates the
following 95% confidence interval: (118.74, 128.86). Her colleague
wants to test (at  = 0.05) whether or not the mean yield for the
population is different from 120 bushels per acre. Based on the
given confidence interval, what can the colleague conclude?
a)
b)
c)
d)
The mean yield is different from 120 and it is statistically significant.
The mean yield is not different from 120 and it is statistically
significant.
The mean yield is different from 120 and it is not statistically
significant.
The mean yield is not different from 120 and it is not statistically
significant.
Using CIs to test (answer)
A researcher is interested in estimating the mean yield (in bushels per
acre) of a variety of corn. From her sample, she calculates the
following 95% confidence interval: (118.74, 128.86). Her colleague
wants to test (at  = 0.05) whether or not the mean yield for the
population is different from 120 bushels per acre. Based on the
given confidence interval, what can the colleague conclude?
a)
b)
c)
d)
The mean yield is different from 120 and it is statistically significant.
The mean yield is not different from 120 and it is statistically
significant.
The mean yield is different from 120 and it is not statistically
significant.
The mean yield is not different from 120 and it is not
statistically significant.