Hypothesis Testing For a Single Population Mean

Example: Grade inflation?

Population of 5 million college

studentsIs the average GPA 2.7?

Sample of 100 college students

How likely is it that 100 students would have an average GPA as large as 2.9 if the population average was 2.7?

The p-value illustrated

How likely is it that 100 students would have an average GPA as large as 2.9 if the population average was 2.7?

Determining the p-value

H0: μ = average population GPA = 2.7HA: μ = average population GPA > 2.7

If 100 students have average GPA of 2.9 with standard deviation of 0.6, the P-value is:

0004.0]33.3[)]100/6.0/()7.29.2([)9.2(

ZPZPXP

Making the decision

• The p-value is “small.” It is unlikely that we would get a sample as large as 2.9 if the average GPA of the population was 2.7.

• Reject H0. There is sufficient evidence to conclude that the average GPA is greater than 2.7.

Terminology

• H0: μ = 2.7 versus HA: μ > 2.7 is called a “right-tailed” or a “one-sided” hypothesis test, since the p-value is in the right tail.

• Z = 3.33 is called the “test statistic”.• If we think our p-value is small if it is less than

0.05, then the probability that we make a Type I error is 0.05. This is called the “significance level” of the test. We say, α=0.05, where α is “alpha”.

Example: Body Temperature?

Population of many, many adults

Is average adult body temperature 98.6 degrees? Or is it lower?

Sample of 80 adults

Average body temperature of 80 sampled adults is 98.4 degrees.

The p-value illustrated

How likely is it that 80 adults would have an average body temp as small as 98.4 if the pop’n average was 98.6?

Determining the p-value

H0: μ = average pop’n body temp = 98.6HA: μ = average pop’n body temp < 98.6

If 80 adults have average body temp of 98.4 with standard deviation of 0.6, the P-value is:

0014.0]98.2[)]80/6.0/()6.984.98([)4.98(

ZPZPXP

Making the decision

• The p-value is “small.” It is unlikely that we would get a sample as small as 98.4 if the average body temp of the population was 98.6.

• Reject H0. There is sufficient evidence to conclude that the average body temp is smaller than 98.6.

Terminology

• H0: μ = 98.6 versus HA: μ < 98.6 is called a “left-tailed” or a “one-sided” hypothesis test, since the p-value is in the left tail.

• Z = -2.98 is the “test statistic”.

• If we think our p-value is small if it is less than 0.02, then the probability that we make a Type I error is 0.02. That is, significance level α = 0.02.

Example: $ on Alcohol?

Population of Penn State students

Is average amount spent weekly $20?

Sample of 64 students

Average amount spent is $17 with standard deviation of $16.

The p-value illustrated

How likely is it that 64 students would spend an average as small as $17, or as large as $23, if the pop’n avg was $20?

Determining the p-value

H0: μ = average $ spent = $20HA: μ = average $ spent $20

If 64 students spend an average of $17 with standard deviation of $16, the P-value is:

067.0]5.1[)]64/16/()2017([)17(

ZPZPXP

067.0)23( XPand

So P-value = 0.067 2 = 0.134

Making the decision

• The p-value is not “small.” It is likely that we would get a sample as small as $17, or as large as $23, if the average amount spent on alcohol was $20.

• Do not reject H0. There is not enough evidence to conclude that the average amount spent differs from $20.

Terminology

• H0: μ = $20 versus HA: μ $20 is called a “two-tailed” or a “two-sided” hypothesis test, since the p-value is in both tails.

• Z = -1.5 is the “test statistic”.

• Since we failed to reject the null hypothesis, we may have made a Type II error.

Very Important Point

• Your p-value will not be correct unless the assumptions are correct!!!!

• You must have a large sample to use the methods presented here.