1. A Machine is expected to produce nails of length 2cm. A random sample of 25 nails gave 
an average length of 2.1cm with standard deviation 0.25cm. Can it be said that the machine is 
producing nails as per specifications? 

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.

Explore hypothesis testing to determine if a machine producing nails meets specifications. Includes a detailed step-by-step one-sample t-test calculation with sample mean 2.1 cm, standard deviation 0.25 cm, and sample size 25.

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