confidence intervals for a population mean€¦ · Øsample size calculations confidence intervals...

Ø Point Estimate

Ø Calculating a Confidence Interval

Ø Interpreting a Confidence Interval

ØAccuracy of Confidence Intervals

Ø Sample Size Calculations

Confidence Intervals for a Population Mean

Lecture 18

Section 6.1

Four Stages of Statistics

• Data Collection þ

• Displaying and Summarizing Data þ

• Probability þ

• Inference• One Quantitative• Confidence Intervals

• Hypothesis Testing

• Inference for a Single Population Mean

• One Categorical

• One Quantitative and One Categorical

• Two Categorical

• Two Quantitative

Motivation: Confidence Intervals

• Scenario: SAT is designed to have normally distributed scores with mean 1000 and standard deviation 195.

• Question: How can we determine if 1000 is a plausible value for the average SAT score of Pitt students?

• Solution #1: Take a ______________________ of SAT scores from Pitt students, find the ________________, and see if it _________________• Problem #1: Cannot sample the ____________________ so we will not find ___

• Problem #2: ________________ exists from sample to sample so the sample mean !̅ will not be a ____________________________ of the population mean

• Problem #3: Cannot display the effect of taking _______________________

• Solution #2: Find an _________________________________ for what the mean SAT score of Pitt students _____________

Types of Estimates

• Point Estimate: a single value that is provided as the estimate of an unknown parameter in a population

• !̅ is a point estimate for "

• # is a point estimate for $

• &̂ is a point estimate for &

• Interval Estimate: an interval of plausible values for an unknown parameter; based on the sample, each value could reasonably be the value of the parameter

Example: Point Estimate

• Scenario: Take a random sample of 25 Pitt students and find a sample mean SAT score of 1250.

• Question: What notation and value should be used to represent a point estimate of the average SAT score of Pitt students?

• Answer: ______________________

• Question: What notation and value should be used to represent the true population mean SAT score of Pitt students?

• Answer: ____ à ___________________________

Example: Point Estimate


• Question: How certain are we that 1250 is the exact population mean SAT score for Pitt students?

• Answer: ________________________________• 1250 probably _________________________________________

• Solution: Use what we know about the _____________ in conjunction with the __________ to calculate an _____________________________

Confidence Interval

• Confidence Interval: interval of plausible values for an unknown parameter that is calculated from the responses in a sample• Provides us with a range of values that could be the true parameter

• Confidence Level: measure of how certain we are that the confidence interval contains the true population parameter• Denoted by 100 1 − ( % where ( is the total area being left out

•Most confidence intervals have the form:

Statistic ± Critical Value ∗ Standard Error

Depends on

confidence levelMargin of Error: maximum expected

difference between statistic and parameter

Point

Estimate

Example: Confidence Interval


• Goal: Calculate a 95% confidence interval for the true mean SAT score of Pitt students.

• Question: What values do we need to calculate the confidence interval?

• Answer:• Statistic: _____________

• Standard Error: _______________________

• Critical Value: ___________ bounding the _______________ of standard normal distribution


• Question: How can we find the Z-scores bound the middle 95% of the standard normal distribution?

• Answer: _______________________________________________• Leaves out a total of ______

Note: Symmetry gives us the

upper critical value of ______.



• Goal: Calculate a 95% confidence interval for the true mean SAT score of Pitt students.

• Answer: ________________________________• Lower Bound:

____________________________________ ___________________________________________

Upper Bound:


• Scenario: Take a random sample of 25 Pitt students and find a sample mean SAT score of 1250. A 95% confidence interval for the average SAT score of Pitt students is (1173.56, 1326.44).

• Question: Is 1000 a plausible value for the average SAT score for Pitt students?

• Answer: ______• 95% confidence interval _______________________________

• 1000 appears to be ___________ because the entire interval is ____________

Confidence Interval for a Single Population Mean

• To estimate a single unknown population mean " using a confidence interval, use:

!̅ ± / ⁄9 :

$

;

where:• !̅: Sample mean

• / ⁄9 :: Critical value corresponding to confidence level 100 1 − ( %

• $: Population standard deviation

• ;: Sample size

Note: Should check to make sure the shape of the sample mean is normal before

calculating the confidence interval.

Critical Values

• Critical Value: multiplier in a confidence interval that tells how many standard error to extend in each direction from the statistic• A confidence interval for " when $ is known will always use <

• Critical values will come from a different distribution in other situations

• The table below shows critical values for typical confidence levels that use <:

Confidence Level Critical Value

80% 1.282

90% 1.645

95% 1.96

98% 2.326

99% 2.576

Example: 99% Confidence Interval

• Scenario: A random sample of the heights of 21 men found a sample mean of 70.6 inches. Assume the population standard deviation is 4 inches.

• Task: Calculate a 99% confidence interval for the mean height of males.

• Step #1: ______________________________• Histogram: _________________

• Skewness: _________________________________, but __________________

• Kurtosis: ___________________________________



• Task: Calculate a 99% confidence interval for the mean height of males.

• Step #2: ____________________________________

__________________________________________________________________________

Note: Even though the sample standard deviation was only 3.363, we still

use 4 because __________________________. It is always better to use the value

of parameters if they are __________.



• Question: How should the confidence interval be interpreted in the context of the situation?

• Answer: ________________________________________________________________ ___________________________________________________________________________• Interpretation of any confidence interval always includes:• __________________________

• ______________________________________________________

• _________________________________________

Example: Measuring Accuracy

• Scenario: IQ scores are known to be normally distributed with a population mean of 100 and a population standard deviation of 15. Take 10 random samples of size 25 and calculate a 90% confidence interval for each.

• Question: How many of these intervals would we expect to contain 100?

• Answer: _____• Confidence intervals are ___________________

• 90% confidence literally means 90% of the time the interval _____________ _______________________, but 10% of the time ____________________

• Confidence levels can always be interpreted as ___________________

Example: Measuring Accuracy

• Scenario: IQ scores are known to be normally distributed with a population mean of 100 and a population standard deviation of 15. Take 10 random samples of size 25 and calculate a 90% confidence interval for each.

• Observations:• 9 of the samples resulted in

__________________________________

• One was a _____________ (__) and __________________________________

Example: Narrower Intervals

• Scenario: A 95% confidence interval for the mean SAT score for Pitt students was (1173.56, 1326.44).

• Question: Keeping all other statistics the same, which of the following will result in a narrower confidence interval?

• Choices:I. Using 90% confidence

II. Using a population standard deviation of 300

III. Using a sample size of 50

• Answer: _________________________I. Critical value would have been ______________

II. More variability in population creates ____________________ in the interval

III. More information helps create a _____________________ interval estimate

Factors Leading to Narrower Intervals

• Question: How can a confidence interval for the population mean become narrower?

!̅ ± / ⁄9 :

$

;

• Answer:• Decrease confidence level

• Decrease standard deviation

• Increase sample size

• Question: Why would we want a narrower interval?

• Answer: Provides a _____________________________ of the parameter• Note: Making an interval narrower by reducing the confidence level

comes with the drawback of __________________________.

Sample Size Calculation

• In many fields, it is known ahead of time how wide a confidence interval is allowed to be. Researchers may need to know how large a sample is necessary to attain that width.

• Given a level of confidence 100 1 − ( % and a population standard deviation $, the sample size necessary to attain a margin of error = is:

; =/ ⁄9 :$

=

:

• Round up to next largest integer if ; is a decimal

Reminder: The margin of error is half the width of the confidence interval.

Example: Sample Size Calculation


• Question: How large a sample would be needed to reduce the width of the 99% confidence interval to 1.5 inches?

70.668.35 72.85

_______ _______70.6

Width: _____ in. Margin of Error: _____ in.

Original Interval

Width: 4.5 inches

Sample Size: 21

New Interval

Width: 1.5 inches

Sample Size: ?

Example: Sample Size Calculation


• Question: How large a sample would be needed to reduce the width of the 99% confidence interval to 1.5 inches?

• Answer:• Margin of Error: ______________________

• Critical Value: ______ confidence à ___________________

• Sample Size: ______________________________________________________

• Round up to _______ people

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