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Lecture 10
Hypothesis Testing
A hypothesis is a conjecture about the distribution of some random variables. For example, a claim about the value of a parameter of the statistical model. There are two types of hypotheses:
The null hypothesis, , is the current belief.
The alternative hypothesis, , is your belief; it is what you
want to show.
Examples: Each of the following situations requires a significance test about a population mean. State the appropriate null hypothesis and alternative hypothesis in each case.
(a) The mean area of the several thousand apartments in a new development is advertised to be 1250 square feet. A tenant group thinks that the apartments are smaller than advertised. They hire an engineer to measure a sample of apartments to test their suspicion.
(b) Larry's car consume on average 32 miles per gallon on the highway. He now switches to a new motor oil that is advertised as increasing gas mileage. After driving 3000 highway miles with the new oil, he wants to determine if his gas mileage actually has increased.
(c) The diameter of a spindle in a small motor is supposed to be 5 millimeters. If the spindle is either too small or too large, the motor will not perform properly. The manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has moved away from the target.
Test Statistic
• The test is based on a statistic that estimates the parameter
that appears in the hypotheses. Usually this is the same
estimate we would use in a confidence interval for the
parameter. When is true, we expect the estimate to take a
value near the parameter value specified in.
- Values of the estimate far from the parameter value specified by give evidence against. The alternative hypothesis determines which directions count against.
- A test statistic measures compatibility between the null hypothesis and the data.
- To assess how far the estimate is from the parameter, standardize the estimate. In many common situations the test statistics has the form
Example: An air freight company wishes to test whether or not the mean weight of parcels shipped on a particular root exceeds 10 pounds. A random sample of 49 shipping orders was examined and found to have average weight of 11 pounds. Assume that the standard deviation of the weights is 2.8 pounds.
Solution:
Example: The hourly wages in a particular industry are normally distributed with mean $13.20 and standard deviation $2.50. A company employs 40 workers paying them an average of $12. per hour. Can this company be accused of paying substandard wages? Use.
Solution:
Decision Errors
When we perform a statistical test we hope that our decision will be correct, but sometimes it will be wrong. There are two possible errors that can be made in hypothesis test.
Definition: The error made by rejecting the null hypothesis when in fact is true is called a type I error. The error made by failing to reject the null hypothesis when in fact is false is called a type II error.
Note: The level of significance of the test is also the probability of type I error, denoted by , i.e.
The probability of a type II error is denoted by.
Example: An experimenter has prepared a drug dosage level that she claims will induce sleep for 80% of people suffering from insomnia. In an attempt to disprove her claim, we administer her prescribed dosage to 20 insomniacs and observe X , the number of people for whom the drug dose induces sleep. We wish to test vs. Assume.
Example: 85% of the general public is right-handed. A survey of 300 chief executive officers of large corporations found that 95% were right-handed. Is this difference in percentages statistically significant? Use. Find the P-value for the test.
Solution:
Tests for a Population Mean ( is known)
where is the specified value of.
Example: In 1999, it was reported that the mean serum cholesterol level for female undergraduates was 168 mg/dl with a standard deviation of 27 mg/dl. A recent study at Baylor University investigated the lipid levels in a cohort of sedentary university students. The mean total cholesterol level among n = 71 females was ̅. Is this evidence that cholesterol levels of sedentary students differ from the previously reported average?
Solution:
Tests for a Population Mean ( is unknown)
Example: Suppose that, for the U.S. data in example before we want to test whether the U.S. average is different from the reported U.K. average.
How to increase the power?
