Question

the diameter of a shaft is assembled on a specific component of a space shuttle is made of titanium and must have 97.5 psi tensile strength. from previous quality inspection, it is known that the standard deviation of tensile strength is 10.4 psi. quality engineers took a sample of 25 shafts and measured the tensile strength. the sample average was x=103psi. quality engineers want to be certain that the shafts will have at least 97.5 psi tensile strength. prepare a hypothesis as well as test statistic and P-Value

Answer 1

The hypothesis test conducted suggests that the population mean tensile strength of the titanium shafts is greater than 97.5 psi, based on the sample data. With a test statistic of 2.644 and a sufficiently low p-value, the null hypothesis can be rejected, indicating the shafts meet the strength requirement.

Step-by-step explanation:

To test the hypothesis that the shafts have at least 97.5 psi tensile strength, we set up the following null (H0) and alternative (Ha) hypotheses:

• H0: μ = 97.5 psi (The population mean tensile strength is equal to 97.5 psi)
• Ha: μ > 97.5 psi (The population mean tensile strength is greater than 97.5 psi)

We will use a one-sample t-test since the population standard deviation is known and the sample size is small (n=25). The test statistic is calculated as follows:

t = (x - μ) / (s/√n)

Plugging in the values, we get:

t = (103 - 97.5) / (10.4/√25) = 5.5 / (10.4/5) = 2.644

Next, we calculate the p-value using a t-distribution table or software for a one-tailed test. Given the high value of t and a degree of freedom (df) of 24, the p-value will be very low, much below the alpha level of 0.05, indicating strong evidence against the null hypothesis.