Conditions Required for a Valid Large-Sample

Confidence Interval for µ

1. A random sample is selected from the target population.

2. The sample size n is large (i.e., n ≥ 30). Due to the Central Limit Theorem, this condition guarantees that the sampling distribution of is approximately normal. Also, for large n, s will be a good estimator of .

x

Thinking Challenge

• We have a random sample of customer order totals with an average of $78.25 and a population standard deviation of $22.5.

• A) Calculate a 90% confidence interval for the mean given a sample size of 40 orders.

• B) Calculate a 90% confidence interval for the mean given a sample size of 75 orders.

• C) Explain the difference in the 90% confidence intervals calculated in A and B.

• D)Calculate the minimum sample size needed to identify a 90% confidence interval for the mean assuming a $5 margin of error.

6.3

Confidence Interval for a Population Mean:

Student’s t-Statistic

Small sample size problem for inference about

• The use of a small sample in making inference about presents two problems when we attempt to use the standard normal z as a test statistic.

Problem 1

• The shape of the sampling distribution of the sample mean now depends on the shape of the population sampled.

• We can no longer assume that the sampling distribution of sample mean is approximately normal because the central limit theorem ensures normality only for samples that are sufficiently large.

Solution to Problem 1

• We know that if our sample comes from a population with normal distribution the sampling distribution of sample mean will be normal regardless of the sample size.

Problem 2

• The population standard deviation is almost always unknown. For small samples the sample standard deviaiton s provides poor approximation for .

Solution to Problem 2(Small Sample with known)

Use the standard normal statistic

z x µ x

x µ n

Solution to Problem 2(Small Sample with Unknown)

Instead of using the standard normal statistic

use the t–statistic

z x µ x

x µ n

t x µs n

in which the sample standard deviation, s, replaces the population standard deviation, .

Student’s t-StatisticThe t-statistic has a sampling distribution very much like that of the z-statistic: mound-shaped, symmetric, with mean 0.

The primary difference between the sampling distributions of t and z is that the t-statistic is more variable than the z-statistic.

Degrees of Freedom

The actual amount of variability in the sampling distribution of t depends on the sample size n. A convenient way of expressing this dependence is to say that the t-statistic has (n – 1) degrees of freedom (df).

zt

Student’s t Distribution

0

t (df = 5)

Standard Normal

t (df = 13)Bell-ShapedSymmetric‘Fatter’ Tails

The smaller the degrees of freedom for t-statistic, the more variable will be its sampling distribution.

• We have a random sample of 15 cars of the same model. Assume that the gas milage for the population is normally distributed with a standard deviaition of 5.2 miles per galon.

• A) Identify the bounds for a 90% confidence interval for the mean given a sample mean of 22.8 miles per gallon.

• B) The car manufacturer of this particular model claims that the average gas milage is 26 miles per gallon. Discuss the validity of this claim using the 90% confidence interval calculated in A.

• C) Let a and b represent the lower and upper boundaries of 90% confidence intervl for the mean of the population. Is it correct to conclude that tere is a 90% probability that true population mean lies between a and b?

Thinking Challenge

• In 1882 Michelson measured the speed of light. His values in km/sec and 299,000 substracted from them. He reported the results of 23 trials with a mean of 756.22 and a standard deviaition of 107.12.

• Find a 95% confidence interval for the true spped of light from these statistics.

• Interpret your result.