Question

presented a sample of \({\rm{n = 153}}\)observations on ultimate tensile strength, the previous section gave summary quantities and requested a large-sample confidence interval. Because the sample size is large, no assumptions about the population distribution are required for the validity of the CI.

a. Is any assumption about the tensile-strength distribution required prior to calculating a lower prediction bound for the tensile strength of the next specimen selected using the method described in this section? Explain.

b. Use a statistical software package to investigate the plausibility of a normal population distribution.

c. Calculate a lower prediction bound with a prediction level of \({\rm{95\% }}\)for the ultimate tensile strength of the next specimen selected.

Short answer

a) Assume normality

b) The population distribution is not normal

c) The lower prediction bound is \(127.7689.\)

Step-by-step solution

To investigate the plausibility of a normal population distribution

(a):

Yes. The assumption that the sample is from the normally distributed population is needed.

Hence Assume normality.

(b):

The normal probability plot suggests that the population distribution is not normal. The indicator of this is that the data does not lay on the line.

Hence the population distribution is not normal

To calculate a lower prediction bound

(c):

The mean of the sample has been calculated and it is

\({\rm{\bar x = 135}}{\rm{.39}}\)

and the sample standard deviation

\({\rm{s = 4}}{\rm{.59}}\)

where \({\rm{n = 153}}\). The t value, for \({\rm{\alpha = 0}}{\rm{.05}}\)which was obtained from

\({\rm{100(1 - \alpha ) = 95}}\)is

\({{\rm{t}}_{{\rm{\alpha ,n - 1}}}}{\rm{ = }}{{\rm{t}}_{{\rm{0}}{\rm{.05,152}}}}{\rm{ = 1}}{\rm{.655}}\)

where the t value was computed using a computer software.

The lower prediction bound can be computed using formula

\({\rm{\bar x - }}{{\rm{t}}_{{\rm{\alpha /2,n - 1}}}}{\rm{ \times s}}\sqrt {{\rm{1 + }}\frac{{\rm{1}}}{{\rm{n}}}} \)

Therefore, the lower prediction bound is

\({\rm{135}}{\rm{.39 - 1}}{\rm{.655 \times 4}}{\rm{.59 \times }}\sqrt {{\rm{1 + }}\frac{{\rm{1}}}{{{\rm{153}}}}} {\rm{ = 127}}{\rm{.7689}}\)