12 bolts are tested for hardness

Question

I am completely lost on this question and on this section in general can you help me with this question??

 

When 12 bolts are tested for hardness, their indexes have a standard deviation of 41.7. Test the claim that the standard deviation of the hardness indexes for all such bolts is greater than 30.0. Use a 0.025 level of significance.

Answer 1

This problem is asking you to test a hypothesis about a standard deviation. In particular, that the standard deviation of the hardness of some bolts is greater than 30.0. As a mathematical statement, this is

σ > 30.0

 

Your null hypothesis (H₀) always contains equality, so this statement is your alternate hypothesis (H₁ or H_A).

H₀: σ = 30.0

H₁: σ > 30.0

 

The information you are given can be summarized as:
n = 12
s = 41.7

Because the inequality is greater than, this is a right (-->) tailed test. You're testing it at a significance level of α = 0.025, so the area to the right of your critical value is 0.025. Since you're testing a standard deviation, you'll be using the χ² distribution. You should have a chart of χ² critical values. The degrees of freedom for a single variable test like this is n - 1 = 11. From the chart, this gives a critical value χ²_R = 21.920.

 

The test statistic for a single variable standard deviation test is

χ² = (n-1)s² / σ²

 

n and s come from the sample data, σ comes from the claim.

 

χ² = (12-1)(41.7)² / (30.0)²

χ² = 21.253

 

The test statistic is closer to the center than your critical value. Right tail: 21.253 < 21.920

Because of this, you cannot reject the null hypothesis. Therefore, there is not sufficient evidence to support the claim that the standard deviation of these bolts' hardness index is greater than 30.0.

 

--

 

Note: Since the test statistic and critical value are so close, it is likely that with a higher significance level (and a corresponding higher chance of error), this data would "pass" the test.