Before accepting a large shipment of bolts, the director of an elevator construction project checks the tensile strength of a simple random sample consisting of 20 bolts. She is concerned that the bolts may be counterfeits, which bear the proper markings for this grade of bolt, but are made from inferior materials. For this application, the genuine bolts are known to have a tensile strength that is normally distributed with a mean of 1400 pounds and a standard deviation of 30 pounds. The mean tensile strength for the bolts tested is 1385 pounds. Formulate and carry out a hypothesis test to examine the possibility that the bolts in the shipment might not be genuine.

Z = (score-mean)/SEm

SEm = SD/√n

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.

To examine the possibility that the bolts in the shipment might not be genuine, we can conduct a hypothesis test.

Step 1: State the hypotheses.
- Null hypothesis (H0): The bolts in the shipment are genuine. μ = 1400.
- Alternative hypothesis (Ha): The bolts in the shipment are not genuine. μ < 1400.

Step 2: Set the significance level (α).
The significance level, denoted by α, determines how much evidence we require to reject the null hypothesis. For this case, let's choose α = 0.05.

Step 3: Calculate the test statistic.
To calculate the test statistic, we will use the formula:

z = (x̄ - μ) / (σ / √n)

Where:
x̄ = sample mean (1385)
μ = population mean (1400)
σ = population standard deviation (30)
n = sample size (20)

Substituting these values into the formula, we get:

z = (1385 - 1400) / (30 / √20)
z = -2.316

Step 4: Determine the critical value.
Since the alternative hypothesis is one-sided (μ < 1400), we need to find the z-value corresponding to the significance level of 0.05 using a standard normal distribution table or calculator.
The critical value is -1.645.

Step 5: Compare the test statistic with the critical value.
If the test statistic falls in the critical region (to the left of the critical value), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In this case, -2.316 < -1.645, so the test statistic falls in the critical region.

Step 6: Make a conclusion.
Since the test statistic falls in the critical region, we reject the null hypothesis. Therefore, based on the given sample evidence, there is sufficient evidence to suggest that the bolts in the shipment might not be genuine.

Note: It is important to remember that hypothesis testing provides evidence for or against a hypothesis but does not confirm or prove definitively that the alternative hypothesis is true.