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Confidence Intervals: Estimating a Population Mean, μ

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Want to estimate average age of San Diego college students Don’t want to sample EVERY STUDENT Instead, I could take a sample (e.g., this class) & estimate μ Xbar! This is called a “point estimate”

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The purpose of a CI is to give an “interval estimate” of the population mean, μ with some degree of certainty

e.g., “We are 95% confident that the mean age of all San Diego college students is between 18-24 years old”

• •

• • •

Confidence Intervals

Used when we can only collect a sample from a large population If we had data from the entire population, we could say EXACTLY what the mean value is with 100% confidence A Confidence Interval (CI) has two parts: Size: range of number values (i.e., 7.3-8.4) Confidence Level (i.e., 95%) Trade-off between confidence and size of confidence interval If I want to be 100% confident in my estimate, would my range be very small or very large? If I only need to be 10% confident, my range can be small

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What do CI’s Mean?

CI X z * ( X )

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What confidence intervals DO NOT mean:

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“There is a 95% chance that my CI includes the true population mean” - NOT TRUE! Either it DOES or it DOESN’T!

“I am 95% confident that my CI includes the true population mean” Even better: “If I ran this study 100 times, and made 100 CI’s in this manner, 95 of my 100 CI’s would include the true population mean”

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Confidence Interval Example

z test example: Does the mean GPA of students who take statistics differ from the mean GPA of all college students? Construct a 95% confidence interval, given: μ = 2.7, σ = 0.6, Xbar = 2.85, sample size n = 36.

σXbar = 0.6/√36 = 0.1

μ = 2.7 z* = -1.96

z* = +1.96

Confidence Intervals CI X z * ( X )

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Xbar = 2.85 z = ±1.96 σXbar = 0.6/√36 = 0.1 –Xbar = 2.85 ± 1.96*(0.1) –Xbar = 2.85 ± 0.196 –Xbar = between 2.654 and 3.046

• • • •

So What??? “We are 95% confident that the population mean lies between 2.654 and 3.046.” The confidence interval we constructed based on our sample includes the overall population mean: 2.70 This is another way of saying that this sample and our population mean aren’t significantly different! That is, we shouldn’t reject H0

2.70

2.654

2.85

σXbar = 0.1

3.046

Confidence Interval Formulas z test CI:

CI X z * ( X )

Single Sample t CI:

CI X t * ( s X )

Dependent t CI:

CI D t * ( s D )

Independent t CI:

CI ( X 1 X 2 ) t * (s X

1

X 2

)

Ch. 12 hw •

A sample of 25 statistics students eats a mean of 9 candy bars per month; it is known that the population mean for college students is 8, with a standard deviation of 4. 1a. Construct a 95% confidence interval for the true population mean of statistics students’ candy bar eating. b. Does this CI indicate a significant difference between statistics students and other students (why or why not)? c. Compare & contrast this analysis with a 2-tailed z-test on the same data. 2. Say a different sample from another college yielded a 95% confidence interval of 8.5 ± 1.5. True or false: a. The true population mean is between 7 and 10. b. About 95% of these students sampled scored between 7 and 10. c. There is a 95% chance that this CI contains the true population mean. d. 5% of confidence intervals constructed this way will not contain the true population mean. 3. The college candy salesman wants a more precise confidence interval to better estimate inventory. Name two ways to obtain a more precise CI.

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• • •

•

•

Want to estimate average age of San Diego college students Don’t want to sample EVERY STUDENT Instead, I could take a sample (e.g., this class) & estimate μ Xbar! This is called a “point estimate”

•

The purpose of a CI is to give an “interval estimate” of the population mean, μ with some degree of certainty

e.g., “We are 95% confident that the mean age of all San Diego college students is between 18-24 years old”

• •

• • •

Confidence Intervals

Used when we can only collect a sample from a large population If we had data from the entire population, we could say EXACTLY what the mean value is with 100% confidence A Confidence Interval (CI) has two parts: Size: range of number values (i.e., 7.3-8.4) Confidence Level (i.e., 95%) Trade-off between confidence and size of confidence interval If I want to be 100% confident in my estimate, would my range be very small or very large? If I only need to be 10% confident, my range can be small

• • •

What do CI’s Mean?

CI X z * ( X )

•

• •

What confidence intervals DO NOT mean:

• •

“There is a 95% chance that my CI includes the true population mean” - NOT TRUE! Either it DOES or it DOESN’T!

“I am 95% confident that my CI includes the true population mean” Even better: “If I ran this study 100 times, and made 100 CI’s in this manner, 95 of my 100 CI’s would include the true population mean”

•

Confidence Interval Example

z test example: Does the mean GPA of students who take statistics differ from the mean GPA of all college students? Construct a 95% confidence interval, given: μ = 2.7, σ = 0.6, Xbar = 2.85, sample size n = 36.

σXbar = 0.6/√36 = 0.1

μ = 2.7 z* = -1.96

z* = +1.96

Confidence Intervals CI X z * ( X )

• • •

Xbar = 2.85 z = ±1.96 σXbar = 0.6/√36 = 0.1 –Xbar = 2.85 ± 1.96*(0.1) –Xbar = 2.85 ± 0.196 –Xbar = between 2.654 and 3.046

• • • •

So What??? “We are 95% confident that the population mean lies between 2.654 and 3.046.” The confidence interval we constructed based on our sample includes the overall population mean: 2.70 This is another way of saying that this sample and our population mean aren’t significantly different! That is, we shouldn’t reject H0

2.70

2.654

2.85

σXbar = 0.1

3.046

Confidence Interval Formulas z test CI:

CI X z * ( X )

Single Sample t CI:

CI X t * ( s X )

Dependent t CI:

CI D t * ( s D )

Independent t CI:

CI ( X 1 X 2 ) t * (s X

1

X 2

)

Ch. 12 hw •

A sample of 25 statistics students eats a mean of 9 candy bars per month; it is known that the population mean for college students is 8, with a standard deviation of 4. 1a. Construct a 95% confidence interval for the true population mean of statistics students’ candy bar eating. b. Does this CI indicate a significant difference between statistics students and other students (why or why not)? c. Compare & contrast this analysis with a 2-tailed z-test on the same data. 2. Say a different sample from another college yielded a 95% confidence interval of 8.5 ± 1.5. True or false: a. The true population mean is between 7 and 10. b. About 95% of these students sampled scored between 7 and 10. c. There is a 95% chance that this CI contains the true population mean. d. 5% of confidence intervals constructed this way will not contain the true population mean. 3. The college candy salesman wants a more precise confidence interval to better estimate inventory. Name two ways to obtain a more precise CI.

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