To solve this problem, we need to determine:

1. The test statistic $z$: The formula for $z$-score is: $z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$ where:

   • $\bar{x} = 16.03$ (sample mean),
   • $\mu = 16$ (hypothesized population mean),
   • $\sigma = 0.24$ (population standard deviation),
   • $n = 69$ (sample size).

   Substituting the values: $z = \frac{16.03 - 16}{0.24 / \sqrt{69}}$

2. The p-value: Since this is a one-tailed test (testing whether the mean is greater than 16), the p-value is the area to the right of the calculated $z$-score under the standard normal curve.

3. Decision based on significance level $\alpha = 0.01$: We compare the $p$-value to $\alpha$. If $p < \alpha$, we reject the null hypothesis $H_0: \mu = 16$.

Let me calculate these values.import scipy.stats as stats import math

Given data

x_bar = 16.03 # sample mean mu = 16 # hypothesized population mean sigma = 0.24 # population standard deviation n = 69 # sample size alpha = 0.01 # significance level

Calculate the test statistic z

z = (x_bar - mu) / (sigma / math.sqrt(n))

Calculate the p-value for a one-tailed test

p_value = 1 - stats.norm.cdf(z)

z, p_value(1.0383279828647987, 0.14955869261591692)(1.0383279828647987, 0.14955869261591692)success1. Test statistic $z$: $1.038$
2. p-value: $0.150$

Decision:

Since the $p$-value ($0.150$) is greater than the significance level $\alpha = 0.01$, there is not sufficient evidence to conclude that the mean length of the bolts is greater than 16 centimeters. The null hypothesis cannot be rejected.

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Do you have any further questions or need detailed steps? Here are 5 related questions to explore:

1. How is the $z$-score affected by changes in sample size?
2. What is the relationship between the $p$-value and the level of significance $\alpha$?
3. Can this test be performed with a smaller sample size and still maintain validity?
4. What would the conclusion be if the significance level were $\alpha = 0.05$?
5. How does the population standard deviation influence the outcome of the test?

Tip: Always double-check the direction of the hypothesis (one-tailed or two-tailed) when determining the $p$-value.