2.6 Confidence Intervals and Margins of Error

What you often see in reports about studies…

These results are accurate to within +/- 3.7%, 19 times out of 20.

Margin of error

Confidence level

95% probability that x is somewhere in the range (x – 3.7, x+3.7) Confidence interval

Confidence Intervals• is sample meanx• We don’t usually know population mean,

(mu)

• We can find confidence intervals– Ranges of values likely to be in– E.g., a 95% confidence level has 0.95

probability of containing

Notation

= allowed error, or probability of error

• (1 – ) = confidence level• = z-score for that confidence interval

– E.g. z0.975 is the z-score for a 95% confidence interval

12

z

Note: the proper notation is actually2

z

Confidence Intervals

• A (1 – ) or (1 – ) x 100% confidence interval for , given population standard deviation , sample size n, and sample mean , represents the range of valuesx

1 12 2

x z x zn n

x z What’s this?

Population mean/Sample Means

• So far, mean of sample = mean of population

• Means from different samples of the same population are different

• Sample means have normal distribution

2

N ,Xn

Common confidence levels and their associated z-scores

Confidence Level

Tail size,

z-score,

90% 0.05 1.645

95% 0.025 1.960

99% 0.005 2.576

2

1

2

z

Example 1: Drying Times• A paint manufacturer knows from experience that

drying times for latex paints have a standard deviation of 10.5 min. The manufacturer wants to use the slogan “Dries in T min.” on its advertising. Twenty test areas of equal size are painted and the mean drying time is found to be 75.4 min.

• A) Find a 95% confidence interval for the actual mean drying time of the paint.

• B) What would be a reasonable value for T?

Example 1• A) For a 95% confidence level, the acceptable

probability error is = 5% = 0.05

12

z

(0.975)z

1.9601 1

2 2

x z x zn n

10.5 10.575.4 (1.960) 75.4 (1.960)

20 20

70.8 80.0 The manufacturer can be 95% confident that the actual mean drying time is between 70.8 min and 80.0 min.

• B) It would be reasonable to advertise “Dries in 80 min.”

Margin of Error and Sample Sizes

• Consider the confidence interval width, w– E.g. 70.8 < < 80.0,

– w = 9.2

• Margin of Error = half the confidence interval width

12

E zn

w = 2E

x–the maximum difference between the observed sample mean and the true value of the population mean

• We can use this to calculate the minimum sample size necessary for a given confidence level– Often used in opinion polls and other surveys

• If sample size too large, waste of resources/time/money• If sample size too small, inaccurate results

Sample Size

12

x zn

12

2w zn

Solving for sample size, n:2

12

2z

nw

Notes

• Need to know in advance

• Estimate it by doing a pre-survey/study

• Margin of error decreases as sample size increases, but only to a point

Example 2: ISPs

• We would like to start an Internet Service Provider (ISP) and need to estimate the average Internet usage of households in one week for our business plan and model. How many households must we randomly select to be 90% sure that the sample mean is within 1 minute of the population mean . Assume that a previous survey of household usage has shown = 6.95 minutes.

Example 2

• For a 90% confidence level, the acceptable probability error is = 10% = 0.10

12

z

(0.995)z

1.645

2

12

2z

nw

22(1.645)(6.95)

2

130.7

You would need a sample of about 131 households.

E = 1 min

w = 2E

= 2 min