Testing a claim about a population mean: t-Test

You work for a steel company that produces "lag bolts". One product is a 10-inch lag bolt. A recent re-design of a machine involved in the production of these bolts has the company wondering whether the production still produces 10-inch lag bolts. You have gathered a random sample of bolts that have been produced with this new production process. Conduct an appropriate test, at a 5% significance level $\displaystyle {\left({t}_{{\alpha/{2}}}={t}_{{0.025}}\approx{2.1098}\right)}$, to determine if the overall mean for this process is 10 inches.

Data (lag bolt lengths, measured in inches):

10.01	9.95	10.05	10.12	9.86	9.86	9.97	10.01	9.95
10.00	10.04	9.92	9.90	10.02	9.92	10.07	10.00	9.93

Sample Statistics

$\displaystyle {n}={18}$ $\displaystyle \overline{{x}}={9.97667}$ $\displaystyle {s}={0.07154}$

1. Choose the correct alternate hypothesis.
   
   • $\displaystyle \mu>{10}$
   • $\displaystyle \mu<{10}$
   • $\displaystyle \mu={10}$
   • $\displaystyle \mu\ge{10}$
   • $\displaystyle \mu\ne{10}$
   • $\displaystyle \mu\le{10}$
2. Are the necessary conditions present to carry out this inference procedure? Explain in context.
   
   • No. The sample is not large enough and the histogram is uniform.
   • Our sample was not larger than 30 in size, but a histogram has a reasonably symmetric bell shape. It is stated that the sample was randomly gathered. We assume that this is a small sample of all possible lag bolts (that there are at least 10*18 produced).

Carry out the procedure ("crunch the numbers"):

3. Sample mean: $\displaystyle \overline{{x}}\approx$
4. Standard error of the sample mean: $\displaystyle \sigma_{\overline{{x}}}\approx$
5. Standardized test statistic: $\displaystyle {t}\approx$
6. P-value:
   

7. Decision:
   
   • Fail to reject the null hypothesis.
   • Reject the null hypothesis.
8. Write a conclusion in context.
   
   • Because our p-value is not less than our level of significance (alpha=0.05), we do not have sufficient evidence to reject the null hypothesis. We cannot conclude that the mean production length of these lag bolts differes from 10 inches.
   • Because our p-value is less than our level of significance (alpha=0.05), we have sufficient evidence to reject the null hypothesis. The evidence seems to indicate that the mean production length of these lag bolts differes from 10 inches.