Question

In a metal fabrication process, metal rods are produced to a specified target length of 15 feet. Suppose that the lengths are normally distributed. A quality control specialist collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and a standard deviation of 0.65 feet. Based on this information answer the following questions to determine if the manufacturer is making rods too short.In a metal fabrication process, metal rods are produced to a specified target length of 15 feet. Suppose that the lengths are normally distributed. A quality control specialist collects a random sample of 16 rods and finds the sample mean length to be 14.8 feet and a standard deviation of 0.65 feet. Based on this information answer the following questions to determine if the manufacturer is making rods too short.

a. Write the null and alternative hypotheses to test this claim. (2 pts.)

b. What is the value of the test statistic? You must write down the formula (2 pts.)

c. What is the associated P-value? (1 pt.)

d. State your conclusion and explain in context using alpha = 0.05. (2 Pts.)

e. Create a 95% Confidence Interval for the true mean length of the rods. Explain what the confidence interval means in context. For example..."We are 95% confident..." (3 pts.)

f. Create a 95% Confidence Interval for the population Standard Deviation of the rods. (2 pts.)

Answer 1

Objective: To determine whether the manufacturer is making rods too short i.e < 15 feet.

Given: For X = random variable that records the length of the metal rod,

Let denote the mean length of the metal rods manufactured.

a. Write the null and alternative hypotheses to test this claim.

To test: Vs

b. What is the value of the test statistic:

The appropriate statistical test to test the above hypothesis, to compare the mean length to a hypothesized value, for unknown population variance, would be a One sample t-test, with test statistic given by:

with the critical region for the left tailed test () given by,

Substituting the values,

c. What is the associated P-value

The range of p-value can be obtained by looking for the t critical values near which the test statistic lies, in the row corresponding to 16 - 1 = 15 degrees of freedom:

We find that the p-value lies between 0.10 < P-value < 0.15. To obtain the exact p-value, we may make use of the excel function:

We get p-value = 0.119

d. State your conclusion and explain in context using alpha = 0.05.

Since the p-value of the test is greater than the fixed significance level, we fail to reject the null hypothesis. We may conclude that the data does not provide sufficient evidence to support the claim that the manufacturer is making the metal rods too short.

e. Create a 95% Confidence Interval for the true mean length of the rods. Explain what the confidence interval means in context.

The 95% CI for mean can be computed using the formula:

Substituting the values, critical value of t for 15 degrees of freedom at 5% level:

Substituting the values:

= (14.454, 15.146)

We are 95% CI that the interval estimate (14.454, 15.146) contains the true population mean.

f. Create a 95% Confidence Interval for the population Standard Deviation of the rods.

From chi-square table:

= (0.231, 1.012)