Hypothesis Testing: IQ Scores & Ball Bearing Diameter Example - Prof. N. Phillips, Exams of Probability and Statistics

Download Hypothesis Testing: IQ Scores & Ball Bearing Diameter Example - Prof. N. Phillips and more Exams Probability and Statistics in PDF only on Docsity!

Math 243: Lecture File 10

N. Christopher Phillips

30 April 2009

A refinement: Significance level

One should really decide ahead of time how small a P-value to ask for. The number is called the significance level and denoted α.

A refinement: Significance level

One should really decide ahead of time how small a P-value to ask for. The number is called the significance level and denoted α.

We have succeeded if, after running our test, it turns out that P ≤ α.

Caution: α is not just another name for P. Instead, it is a criterion we choose before the test, preferably even before the experiment, like what the treatments are.

A refinement: Significance level

One should really decide ahead of time how small a P-value to ask for. The number is called the significance level and denoted α.

We have succeeded if, after running our test, it turns out that P ≤ α.

Caution: α is not just another name for P. Instead, it is a criterion we choose before the test, preferably even before the experiment, like what the treatments are.

Caution: “significant” does not mean important here. Rather, it means the evidence for something is strong.

Recall Example 7

You have bought a large quantity of 7 mm ball bearings from Wang’s Widgets Inc., and you are preparing to sue Wang’s Widgets Inc. because, you claim, the diameter of the balls in the ball bearings is not as advertised.

We compare the mean diameter of the balls in the ball bearings with 7 mm, which is what the diameter is supposed to be.

Let μ be the mean diameter, in mm, of the balls in 7 mm ball bearings from Wang’s Widgets Inc.

H 0 : μ = 7. Ha: μ 6 = 7.

Recall Example 7

You have bought a large quantity of 7 mm ball bearings from Wang’s Widgets Inc., and you are preparing to sue Wang’s Widgets Inc. because, you claim, the diameter of the balls in the ball bearings is not as advertised.

We compare the mean diameter of the balls in the ball bearings with 7 mm, which is what the diameter is supposed to be.

Let μ be the mean diameter, in mm, of the balls in 7 mm ball bearings from Wang’s Widgets Inc.

H 0 : μ = 7. Ha: μ 6 = 7.

Our lawyers are careful, and ask that we run a test at significance α = 0. 01.

Example 7 (continued)

Let μ be the mean diameter, in mm, of the balls in 7 mm ball bearings from Wang’s Widgets Inc. Suppose the standard deviation of their diameters is known to be 0.1 (in mm). H 0 : μ = 7. Ha: μ 6 = 7. We are asking for significance level α = 0. 01.

We choose a simple random sample of size 100, and get x = 7. 03. So

Example 7 (continued)

Let μ be the mean diameter, in mm, of the balls in 7 mm ball bearings from Wang’s Widgets Inc. Suppose the standard deviation of their diameters is known to be 0.1 (in mm). H 0 : μ = 7. Ha: μ 6 = 7. We are asking for significance level α = 0. 01.

We choose a simple random sample of size 100, and get x = 7. 03. So

z = x − μ 0 σ/

n

We got P ≈ 0. 003.

Example 7 (continued)

Let μ be the mean diameter, in mm, of the balls in 7 mm ball bearings from Wang’s Widgets Inc. H 0 : μ = 7. Ha: μ 6 = 7. We are asking for significance level α = 0. 01 , and we got P ≈ 0. 003. Is it true that P ≤ α?

Example 7 (continued)

Let μ be the mean diameter, in mm, of the balls in 7 mm ball bearings from Wang’s Widgets Inc. H 0 : μ = 7. Ha: μ 6 = 7. We are asking for significance level α = 0. 01 , and we got P ≈ 0. 003. Is it true that P ≤ α?

Yes, P ≤ α.

Example 7 (continued)

Let μ be the mean diameter, in mm, of the balls in 7 mm ball bearings from Wang’s Widgets Inc. H 0 : μ = 7. Ha: μ 6 = 7. We are asking for significance level α = 0. 01 , and we got P ≈ 0. 003. Is it true that P ≤ α?

Yes, P ≤ α.

Accordingly, in formal terms, we reject the null hypothesis.

In terms related to the statement of the problem, we conclude that there is strong evidence, at the significance level α = 0. 01 , that the mean diameter of the balls in 7 mm ball bearings from Wang’s Widgets Inc. is not really 7 mm.

Example 7 (continued)

Let μ be the mean diameter, in mm, of the balls in 7 mm ball bearings from Wang’s Widgets Inc. H 0 : μ = 7. Ha: μ 6 = 7. We are asking for significance level α = 0. 01 , and we got P ≤ α.

Accordingly, in formal terms, we reject the null hypothesis.

In terms related to the statement of the problem, we conclude that there is strong evidence, at the significance level α = 0. 01 , that the mean diameter of the balls in 7 mm ball bearings from Wang’s Widgets Inc. is not really 7 mm.

Example 7 (continued)

As before: You have bought a large quantity of 7 mm ball bearings from Wang’s Widgets Inc., and you are preparing to sue Wang’s Widgets Inc. because, you claim, the diameter of the balls in the ball bearings is not as advertised.

We compare the mean diameter of the balls in the ball bearings with 7 mm, which is what the diameter is supposed to be.

Let μ be the mean diameter, in mm, of the balls in 7 mm ball bearings from Wang’s Widgets Inc.

H 0 : μ = 7. Ha: μ 6 = 7.

Our lawyers are careful, and ask that we run a test at significance α = 0. 01.

Example 7 (continued)

Let μ be the mean diameter, in mm, of the balls in 7 mm ball bearings from Wang’s Widgets Inc. Suppose the standard deviation of their diameters is known to be 0.1 (in mm). H 0 : μ = 7. Ha: μ 6 = 7. We are asking for significance level α = 0. 01. We choose a simple random sample of size 100. Suppose that we instead get x = 6. 98. So