Download STA220 – Guided Notes 6.3

STA220 – Guided Notes 6.3
To test hypotheses regarding the population mean assuming the population standard deviation
is known, two requirements must be satisfied:
1. A simple random sample is obtained.
2. The population from which the sample is drawn is normally distributed or the sample
size is large (n≥30).
Example 1:
A trucking company claims that the average weight of a fully loaded moving van is 12,000 lb. The highway patrol
decides to check this claim. A random sample of 30 moving vans shows that the average weight is 12,100 lb. with a
standard deviation of 800 lb. Construct a hypothesis test to determine whether the average weight of a moving
van is more than 12,000 lb. Use a 5% level of significance.
Example 2:
Jerry is doing a project for his sociology class in which he tests the claim that the Pleasant View housing project
contains family units of average size 3.3 people (the national average). A random sample of 64 families from
Pleasant View project shows a sample mean of 3.8 people per family unit with sample standard deviation 1.3.
Construct a hypothesis test to determine whether the average size of a family unit in Pleasant View is different
from the national average of 3.3. Use a 2% level of significance.
The observed significance level, or p-value, for a test is the probability of observing the results
actually observed (z*) assuming the null hypothesis is true.
𝑃(𝑧 ≥ 𝑧 ∗ |𝐻𝑜 )
The lower this probability, the less likely H0 is true.
Procedure
Example 3
In a test 𝐻0 : 𝜇 = 100 against 𝐻𝑎 : 𝜇 ≠ 100, the sample data yielded the test statistic z = 2.32. Find the p-value for
the test.
Example 4:
The amount of time to finish the US census is of interest to the federal government. A member of the Census
bureau claims it takes no more than ten minutes to fill out the census. A sample of 52 randomly chosen citizens
were timed while completing the census. They had a mean of 10.6 minutes and a standard deviation of 2.25
minutes. Use a 5% significance level and the p-value method to test the claim from the member of the census
bureau.
Example 5:
Historically, the average height for males was believed to be 68 inches. A doctor believes the average height has
increased over the last 100 years. He claims the average male is now 70 inches tall. A random sample of 50 men
had an average height of 68.9 inches and a standard deviation of 2.8 inches. Use a 10% significance level and the pvalue method to test the doctor’s claim.
Test Hypothesis about a Population Mean: t-Statistic
Procedure
Definition
Example 6:
The lifespan for the general population of males born in 1980 is 77 years old. A worker for the Census Bureau
claims that the average lifespan for college professors is greater than 77. A random sample of 17 deceased college
professors had a mean lifespan of 89 and a standard deviation of 9.5 years. Use a significance level of 10% to test
the CB worker’s claim.
Example 7:
The Natural Foods Diet claims that people lose an average of ten pounds in two months on the plan. A random
sample of 26 people lost an average 8.9 pounds on the diet in two months. The standard deviation was 3.25
pounds. Use a 2% significance level to test the claim that the diet helps people lose an average of 10 pounds in two
months.