The goal of inferential statistics is to use a sample to make an inference about a population.

A class of 50 students wants to study the average GPA at KSU.

Student number 1 collects a sample of 5 student GPA's.

S₁={3.01, 3.28, 2.97, 3.41, 3.21}

=3.176

Student number 2 collects a sample of 5 student GPA's.

S₂={2.89, 3.33, 1.97, 2.59, 3.01}

=2.758

Student number 3 collects a sample of 5 student GPA's.

S₃={2.93, 2.78, 3.41, 3.17, 2.81}

=3.02

The remaining 47 students proceed in a similar fashion.

Are there differences in the variations in the single observations and the variations of the sample averages?

Given 50 sample averages what might you do to estimate the true population average?

The sampling distribution of a sample statistic is the distribution of the values of the statistic created by repeated samples of n observations.

The average of the sampling distribution of the averages, , is the same as the population average m. That is.

The standard error of the sampling distribution of the average, , is smaller than the standard deviation of the population. Specifically, .

If the sample size is sufficiently large, then the mean of a random sample from a population has a sampling distribution that is approximately normal, regardless of the shape of the distribution of the population. As the sample size increases, the better the approximation will be.

example: The average GPA at a particular school is m=2.89 with a standard deviation s=0.63. A random sample of 25 students is collected. Find the probability that the average GPA for this sample is greater than 3.0.

The average is standard error is

The z-score is . Looking up this z-score in the normal curve table yields a probability of .8078. The final answer is 1-.8078=.1922.

example: The time it takes students in a cooking school to learn to prepare seafood gumbo is a random variable with a normal distribution where the average is 3.2 hours with a standard deviation of 1.8 hours.

i. Find the probability that the average time it will take a class of 36 students to learn to prepare seafood gumbo is less than 3.4 hours.

ii. Find the probability that it takes one student between 3 and 4 hours to learn to prepare seafood gumbo.

 

 

Homework

Section 5-5:  1-5, 8, 9, 11, 13, 16