Question

1) An engineer who is studying the tensile strength of a steel alloy knows that tensile strength
is approximately normally distributed with σ = 60 psi. A random sample of 12 specimens
has a mean tensile strength of 3450 psi.

a) Compute a two-sided confidence interval on the mean tensile strength at a 95%
confidence level

b) Test the hypothesis that the mean strength is 3500 psi at α= 0.05, and provide a
conclusion statement

c) If we keep the significance level constant (alpha). What would you recommend to make
the interval smaller?

d) How would the interval change if we did not know that σ = 60 psi, but the number of
specimens is 80?

How would the interval change if we did not know that σ = 60 psi, but the number of
specimens is 10?

f) Explain if your answers in a) and b) agree with each other.

Answer 1

(a)

= Sample Mean = 3450

= Population Standard Deviation = 60

n = Sample Size = 12

=0.05

From Table, critical values of Z = 1.96

Confidence Interval:

So,

Answer is:

(3416.052, 3483.948)

(b)

H0: Null Hypothesis: = 3500 ( the mean strength is 3500 psi)

HA: Alternative Hypothesis: 3500 ( the mean strength is not 3500 psi)

= Sample Mean = 3450

= Population Standard Deviation = 60

n = Sample Size = 12

=0.05

From Table, critical values of Z = 1.96

Test Staistic is given by:

Since calculated value of Z = - 2.887 is less than critical value of Z = - 1.96, the difference is significant. Reject null hypothesis.

Conclusion:

The data do not support the claim that the mean strength is 3500 psi.

(c)

If we keep the significance level constant (alpha). we recommend increase the sample size to make
the interval smaller.

(d)

The confidence interval will not change if we did not know that σ = 60 psi, but the number of
specimens is 80 because by Central limit theorem, for sample size = n = 80 > 30, large sample, Z - distribution is still applicable.

(e)

the confidence interval will be narrower if we did not know that σ = 60 psi, but the number of
specimens is 10 , because by Central limit theorem, for sample size = n = 10 < 30, small sample, only t - distribution is applicable.

(f)

Our answers in a) and b) agree with each other.

Explanation:
Since all values in the Confidence Interval (3416.052, 3483.948) are less than 3500, we conclude that the data do not support the claim that the mean strength is 3500 psi.