Question

The diameter of steel rods manufactured on two different extrusion machines is being investigated. Two random samples of sizes ?1 = 20 and ?2 = 24 are selected, and the sample means and sample variances are ?̅̅1̅ = 9.22, ?1 = 0.55, ?̅̅2̅ = 9.43, and ?2 = 0.62, respectively. Assume that ?1 2 ≠ ?2 2 and that the data are drawn from a normal distribution. Is there evidence to support the claim that machine 1 produces rods with smaller mean diameters than machine 2? Use ? = 0.05.

a. Write the appropriate hypothesis.

b. Use P-value approach for hypothesis testing

. c. Use t-test for hypothesis testing.

d. Use confidence interval for hypothesis testing.

e. Clearly write your conclusion.

Answer 1

The provided information is:

Sample 1:

Sample 2:

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(a). The investigator wants to test the claim that machine 1 produces rods with smaller mean diameters than machine 2 at the significance level 0.05.

Let represents the true mean diameter of machine 1.

Let represents the true mean diameter of machine 2.

The null and the alternative hypothesis formulated as,

  

The test is left tailed.

(b).

If the obtained p-value is less than the alpha value, then null hypothesis gets rejected.

If the obtained p-value is greater than the alpha value, then null hypothesis does not rejected.

(c).

The test statistic is given as,

  

The degrees of freedom is given by,

\\

At the degrees of freedom 42 and the test statistic (-1.19), the p-value is obtained by using the t-table as,

Therefore, the p-values is 0.1204

Since the p-value (0.1204) is greater than significance level (0.05), so the null hypothesis does not rejected.

(d).

At the degrees of freedom 42 and the alpha = 0.05, the two-tailed critical value obtained by using the t-table is +/- 2.0181.

The 95% confidence interval for the difference of means is given by,

Since the null value 0 lies within the estimated confidence interval limits (-0.5661, 0.1461), so the researcher does not reject the null hypothesis.

(e).

Since the null hypothesis does not gets rejected, so it can be concluded that there is an insufficient evidence to support the claim that machine 1 produces rods with smaller mean diameters than machine 2 at the significance level 0.05.