How confident are we that our sample means make sense?

Confidence intervals

Maybe not

Confidence intervals

1. Find a confidence interval for the population mean using a sample from a normal distribution with known variance.

2. Find a confidence interval for the population mean using a sample from any distribution with known or unknown variance

3. Make inference from confidence intervals.

Point estimate

• A single number that estimates a population parameter is called a point estimate

How confident we are about an estimate depends on two factors:

• The size of the sample – the larger the size of the sample, the closer the estimate is likely to be to the true population mean

• The variance of the population – if readings are generally more varied the estimate will be less reliable

From previous lesson• :

standard error

where is the populationstandard deviation

n

When comparing two estimates of the same or a similar parameters.

Point estimateGiven two samples:1. Get each of their unbiased estimator of mean

(point estimate)

2. Calculate their standard errors

3. Rule of thumb the one with the smallest standard error, is more likely to have an estimated mean nearer to the population true mean.

Task

• Exercise A (page 111)

Difficulty using point estimate

• In complex situations the choice of an estimate of a population parameter is not always clear. Statisticians may find that they have no idea how to use the sample data to estimate the population parameter of interest, or they may in fact have several equally plausible competing estimates to select from.

• Point estimates do not inform us about how much the estimate is likely to be in error.

NORMAL DISTRIBUTION SAMPLE MEANS

Confidence Interval

6.00

9.00

9090%

100

Confidence Intervals

• A point estimate is the middle point of the interval and the endpoints of the interval communicate the size of the error associated with the estimate and how ‘confident’ we are that the population parameter is in the interval

6.00

9.00

Confidence Interval Calculation• Typical confidence levels used in practice for confidence

intervals are 90%, 95% or 99% with 95% occurring most frequently

• Find 90% from the percentage pointstable p=0.95 and the valuez=1.6449 ≈ 1.645

Confidence interval= 1.645xn

Confidence Interval

The higher the level of confidence, the wider the confidence interval needs to be.

90% and 95% confidence intervals are given by:

90% find p=0.95 z=1.6449 1.64595% find p=0.975 z=1.96

1.645 , 1.645 90% interval

1.96 , 1.96 95% interval

x xn n

x xn n

Task

• Exercise B Page 116

All the questions that you have done so far have been from populations that

have been normally distributed.

CLT states that if sample sizes are large enough then the mean of any distribution is approximately normally distributed with a standard error

therefore, any random sample where n is big enough will have a 95% confidence interval given by:

n

1.96 , 1.96x xn n

Using an estimated variance

• When the σ² is not known but n ≥ 30 an unbiased estimate of the variance S2 can be calculated using

• So when the variance or standard deviation are not known replace σ² with S2 and σ with S

2

22

2

1 1i

xxx x nS

n n

Task

• Read examples 4 & 5• Exercise D page 120

Key points

• A 95% confidence interval tells us that there is a probability of 0.95 that the interval contains the population μ

• If the sample is taken from a normal population then a 95% confidence interval is given by

• If the sample is taken from any distribution and n is large enough, then a 95% confidence interval is given by

• If the population is not know then replace σ² with S2 and σ with S and 95% can by given by:

1.96 , 1.96x xn n

1.96 , 1.96x xn n

1.96 , 1.96S S

x xn n

Task

• Mixed questions – page 122

• Homework or now– Test yourself page 122

Confidence intervals

1. Find a confidence interval for the population mean using a sample from a normal distribution with known variance.

2. Find a confidence interval for the population mean using a sample from any distribution with known or unknown variance

3. Make inference from confidence intervals.