The diameter of a shaft assembled on a specific component of a space shuttle is made of titanium and must have a tensile strength of at least 97.5 psi. From previous quality inspections, 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, finding a sample average of [tex]x = 103 \, \text{psi}[/tex].

Quality engineers want to be certain that the shafts will have at least 97.5 psi tensile strength. Prepare a hypothesis test, including the test statistic and P-value.

1. **State the Hypotheses:**
- Null Hypothesis ([tex]H_0[/tex]): The mean tensile strength is at least 97.5 psi ([tex]\mu \geq 97.5[/tex]).
- Alternative Hypothesis ([tex]H_a[/tex]): The mean tensile strength is less than 97.5 psi ([tex]\mu < 97.5[/tex]).

2. **Test Statistic:**
- Use the formula for the test statistic:
\[
z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}
\]
where:
- [tex]\bar{x} = 103[/tex] psi (sample mean)
- [tex]\mu_0 = 97.5[/tex] psi (hypothesized population mean)
- [tex]\sigma = 10.4[/tex] psi (population standard deviation)
- [tex]n = 25[/tex] (sample size)

3. **P-Value:**
- Calculate the P-value using the test statistic.

Note: The completion of the calculation and interpretation of the P-value will depend on statistical software or Z-table references.