Statistics Confidence Intervals

Statistics Confidence Intervals

Confidence Intervals We will use the sample mean 𝒙 to estimate the unknown population mean µ

Confidence Intervals Using the sample mean 𝒙 to estimate the unknown population mean µ is called “making inferences”

Confidence Intervals If you can assume the distribution of the sample means is normal, you can use the normal distribution probabilities for making probability statements about µ

as “n” increases, variability (spread) also decreases
Confidence Intervals as “n” increases, variability (spread) also decreases

Confidence Intervals We use: s/ n for the measure of variability in the new population of 𝒙 s

Confidence Intervals The standard deviation of the 𝒙 s: s/ n is called the “standard error” abbreviated “se”

Confidence Intervals 𝒙 -3se 𝒙 -2se 𝒙 -se 𝒙 𝒙 +se 𝒙 +2se 𝒙 +3se
So our normal curve for the true value of the population mean µ is: 𝒙 -3se 𝒙 -2se 𝒙 -se 𝒙 𝒙 +se 𝒙 +2se 𝒙 +3se

Confidence Intervals About 95% of the possible values for μ will be within 2 SE of 𝒙

Inferences About μ Although our normal curve graphs the 𝒙 s, we will use it to make inferences about what value μ actually has

This allows us to create a “confidence interval” for values of μ
Confidence Intervals This allows us to create a “confidence interval” for values of μ

Confidence Intervals Confidence interval formula: 𝒙 - 2s/ n ≤ μ ≤ 𝒙 + 2s/ n or 𝒙 - 2se ≤ μ ≤ 𝒙 + 2se With a confidence level of 95%

The “2” in the equations is called the “critical value”
Confidence Intervals The “2” in the equations is called the “critical value”

It comes from the normal curve, which gives us the 95%
Confidence Intervals It comes from the normal curve, which gives us the 95%

2s/ n or 2se is called the “margin of error”
Confidence Intervals 2s/ n or 2se is called the “margin of error”

What if we wanted a confidence level of 99%
CONFIDENCE INTERVALS PROJECT QUESTION What if we wanted a confidence level of 99%

CONFIDENCE INTERVALS PROJECT QUESTION What if we wanted a confidence level of 99% We’d use a value of “3” rather than 2

Confidence Intervals For most scientific purposes, 95% is “good-enuff” In the law, 98% is required for a criminal case In medicine, 99% is required

Confidence Intervals For a 95% confidence interval, 95% of the values of μ will be within 2se of 𝒙

Confidence Intervals If we use the confidence interval to estimate a likely range for true values of μ, we will be right 95% of the time

For a 95% confidence interval, we will be WRONG 5% of the time
Confidence Intervals For a 95% confidence interval, we will be WRONG 5% of the time

For a 99% confidence interval, how much of the time will we be wrong?
CONFIDENCE INTERVALS PROJECT QUESTION For a 99% confidence interval, how much of the time will we be wrong?

CONFIDENCE INTERVALS PROJECT QUESTION For a 99% confidence interval, how much of the time will we be wrong? we will be wrong 1% of the time

Confidence Intervals The percent of time we are willing to be wrong is called “α” (“alpha”) or “the α-level”

Confidence Intervals Everyday use of confidence intervals: You will frequently hear that a poll has a candidate ahead by 10 points with a margin of error of 3 points

Confidence Intervals This means: 10-3 ≤ true difference ≤ 10+3 Or, the true difference is between 7 and 13 points (with 95% likelihood)

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 Can we assume normality?

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 Can we assume normality? yes, because n>20

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the α-level?

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the α-level? 5%

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the critical value?

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the critical value? 2, because we want a 95% confidence interval

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the standard error?

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the standard error? s/ n = 5/ 25 = 5/5 = 1

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the margin of error?

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the margin of error? 2se = 2(1) = 2

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the confidence interval?

CONFIDENCE INTERVALS PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 25 What is the confidence interval? 𝒙 - 2s/ n ≤ μ ≤ 𝒙 + 2s/ n 7 – 2 ≤ μ ≤ ≤ μ ≤ 9 with 95% confidence

CONFIDENCE INTERVALS PROJECT QUESTION Interpreting confidence intervals: If the 95% confidence interval is: 5 ≤ µ ≤ 9 Is it likely that µ = 10?

CONFIDENCE INTERVALS PROJECT QUESTION No, because it’s outside of the interval That would only happen 5% of the time

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 Can we assume normality?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 Can we assume normality? no, because n<20-30

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 Can we assume normality? no, because n<20-30 But if the data is normal-ish, we can!

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the α-level?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the α-level? 5%

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the critical value?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the critical value? 2, because we want a 95% confidence interval

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the standard error?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the standard error? s/ n = 5/ 16 = 5/4 = 1.25

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the margin of error?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the margin of error? 2se = 2(1.25) = 2.5

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the confidence interval?

Confidence Intervals PROJECT QUESTION Find a 95% confidence interval for μ given: 𝒙 = 7 s = 5 n = 16 What is the confidence interval? 𝒙 - 2s/ n ≤ μ ≤ 𝒙 + 2s/ n 7 – 2.5 ≤ μ ≤ ≤ μ ≤ 9.5 with 95% confidence

Confidence Intervals PROJECT QUESTION Interpreting confidence intervals: If the 95% confidence interval is: 4.5 ≤ µ ≤ 9.5 Is it likely that µ = 10?

Confidence Intervals PROJECT QUESTION No, because it’s outside of the interval That would only happen 5% of the time

Find the 95% confidence interval for μ given: 𝒙 = 53 s = 14 n = 49
CONFIDENCE INTERVALS PROJECT QUESTION Find the 95% confidence interval for μ given: 𝒙 = 53 s = 14 n = 49

CONFIDENCE INTERVALS PROJECT QUESTION Find the 95% confidence interval for μ given: 𝒙 = 53 s = 14 n = ≤ µ ≤ 57

Can you say with 95% confidence that µ ≠ 55?
CONFIDENCE INTERVALS PROJECT QUESTION Can you say with 95% confidence that µ ≠ 55?

CONFIDENCE INTERVALS PROJECT QUESTION Can you say with 95% confidence that µ ≠ 55? Nope… it’s in the interval It IS a likely value for µ

Find the 95% confidence interval for μ given: 𝒙 = 481 s = 154 n = 121
CONFIDENCE INTERVALS PROJECT QUESTION Find the 95% confidence interval for μ given: 𝒙 = 481 s = 154 n = 121

CONFIDENCE INTERVALS PROJECT QUESTION Find the 95% confidence interval for μ given: 𝒙 = 481 s = 154 n = ≤ µ ≤ 509

Can you say with 95% confidence that µ might be 450?
CONFIDENCE INTERVALS PROJECT QUESTION Can you say with 95% confidence that µ might be 450?

CONFIDENCE INTERVALS PROJECT QUESTION Can you say with 95% confidence that µ might be 450? µ is unlikely to be 450 – that value is outside of the confidence interval and would only happen 5% of the time

You will have a smaller interval if you have a larger value for n
Confidence Intervals You will have a smaller interval if you have a larger value for n

So you want to take the LARGEST sample you can
Confidence Intervals So you want to take the LARGEST sample you can

This is called the “LAW OF LARGE NUMBERS”
Confidence Intervals This is called the “LAW OF LARGE NUMBERS”

What if you have a sample size smaller than 20???
Confidence Intervals What if you have a sample size smaller than 20???

Confidence Intervals What if you have a sample size smaller than 20???
You must use a different (bigger) critical value W.S. Gosset 1908

Questions?

Confidence Intervals We’ve done confidence intervals for measurement data with a mean μ based on the sample mean 𝒙 and margin of error 2se

CI for Proportions What about count data and proportions?

CI for Proportions We’ll use the normal curve for proportions:
p -3 pq n p -2 pq n p - pq n p p + pq n p +2 pq n p +3 pq n

If p = .4 and n = 30 find the 95% CI for p
CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p

If p = .4 and n = 30 find the 95% CI for p se =
CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p se =

If p = .4 and n = 30 find the 95% CI for p se = pq n = .4x.6 30 ≈ .089
CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p se = pq n = .4x.6 30 ≈ .089

If p = .4 and n = 30 find the 95% CI for p me =
CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p me =

If p = .4 and n = 30 find the 95% CI for p me = 2 × .089 = .178
CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p me = 2 × .089 = .178

If p = .4 and n = 30 find the 95% CI for p CI:
CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p CI:

CI FOR PROPORTIONS PROJECT QUESTION If p = .4 and n = 30 find the 95% CI for p CI: ≤ p ≤ ≤ p ≤ .578

Questions?

Statistics Confidence Intervals

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