Confidence Intervals

Means

Remember: In sampling distribution of means…..

Approximately normal if n is large ( n 30) – CLTThe population is normal

x

x n

So how will I find a confidence intervals for a mean? How is it different from a proportion?

Formula for a z-confidence interval:

2x z

n

Suppose you work for a consumer advocate agency & want to find the mean repair cost of a washing machine. You randomly select 40 repair costs and find the mean to be $100. The standard deviation is $17.50. construct a 95% interval for the population mean.

Find the minimum required sample size if you want to be 95% confident that the sample mean is within 2 units of the population mean if the population standard deviation is 4.8.

But what if the sample size is small or the population standard deviation is unknown?

We use the t-distribution if the population standard deviation is unknown.

It’s bell shaped – centered at 0. Each t-distribution is more spread out than

the z-distribution (normal). As the sample size increases, the spread

decreases and actually approaches the normal distribution.

Based on degrees of freedom (df = n-1)

Degrees of freedom: (n-1)

Suppose A + B + C + D = 18

Thus there are 3 degrees of freedom

Free to be anything

The t- distribution compared to the normal distribution http://www.nku.edu/~longa/stats/taryk/TDist

.html

In normal sampling but with t-distribution

xz

n

x

tsn

Reading the t-chart

The Raman Arches is an Italian restaurant. The manager wants to estimate the average amount a customer spends on lunch. A random sample of 115 customers’ lunch tabs gave a mean of $9.74 with a standard deviation of $2.93. Find and interpret a 90% confidence interval for the average amount spent on lunch by all customers.

A sample of 20 students had a test average of 100 with a standard deviation of 4.2 points. Find and interpret a 90% confidence interval for the average test score for all students on this test.

A random sample of large tents listed in Consumer Reports: Special Outdoor Issue gave the following prices. Find a 90% confidence interval for the mean price of all such tents.

115 110

140 135

80 110

150 210

250 120

230 130

Homework - Worksheet