To determine whether the machine is producing nails as per specifications, we can perform a hypothesis test. Specifically, we'll conduct a one-sample t-test since the population standard deviation is unknown and the sample size is small (n = 25).

Steps:

1. State the Hypotheses:

   • Null Hypothesis (H₀): The machine is producing nails as per specifications, i.e., the mean length of the nails is 2 cm. $\mu = 2$
   • Alternative Hypothesis (H₁): The machine is not producing nails as per specifications, i.e., the mean length of the nails is different from 2 cm. $\mu \neq 2$

2. Select the Significance Level:

   • Typically, a significance level ($\alpha$) of 0.05 is used. This means we are willing to accept a 5% chance of rejecting the null hypothesis when it is actually true.

3. Calculate the Test Statistic:

   • The test statistic for a one-sample t-test is calculated using the formula: $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$ where:
   • $\bar{x}$ = sample mean = 2.1 cm
   • $\mu_0$ = population mean under the null hypothesis = 2 cm
   • $s$ = sample standard deviation = 0.25 cm
   • $n$ = sample size = 25

   Plugging in the values: $t = \frac{2.1 - 2}{0.25/\sqrt{25}} = \frac{0.1}{0.05} = 2.0$

4. Determine the Degrees of Freedom:

   • The degrees of freedom (df) for this test is $n - 1 = 25 - 1 = 24$.

5. Find the Critical t-value:

   • For a two-tailed test with $\alpha = 0.05$ and df = 24, the critical t-value can be found using a t-distribution table or calculator. The critical value for df = 24 at $\alpha/2 = 0.025$ is approximately $\pm 2.064$.

6. Make the Decision:

   • Compare the calculated t-value with the critical t-value.
   • If $|t|$ (the absolute value of the test statistic) is greater than the critical t-value, we reject the null hypothesis.

   In this case, $t = 2.0$ and the critical t-value is 2.064. Since 2.0 is less than 2.064, we fail to reject the null hypothesis.

Conclusion:

There is not enough evidence to conclude that the machine is producing nails that deviate from the specified length of 2 cm. Hence, we can say that the machine is producing nails as per specifications.

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Would you like more details on any part of this process or have any questions? Here are some related questions you might find interesting:

1. How do you interpret the p-value in hypothesis testing?
2. What are the assumptions of a t-test?
3. How would the result change with a larger sample size?
4. What is the difference between a one-sample t-test and a two-sample t-test?
5. How do you perform a t-test if the population standard deviation is known?

Tip: Always ensure that the assumptions of the t-test are satisfied before proceeding with the test, especially the normality of the data when the sample size is small.