In a factory, the following null hypothesis is formulated for its defect levels. H0: mu<=60, H1: mu != 60.  
From an inspection report, the samples showed the following values, n = 16, sd = 12 and  mean = 62. Test the hypothesis


To test the hypothesis, we'll conduct a **two-tailed hypothesis test** using the **t-test**. The steps are as follows:

### Step 1: State the null and alternative hypotheses
- **Null Hypothesis (H₀)**: \(\mu \leq 60\)
- **Alternative Hypothesis (H₁)**: \(\mu \neq 60\)

We are testing whether the sample mean differs significantly from the hypothesized population mean of 60.

### Step 2: Set the significance level
Typically, we use a significance level (\(\alpha\)) of 0.05 unless specified otherwise.

### Step 3: Calculate the test statistic (t-statistic)

We use the formula for the **t-statistic**:
\[
t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
\]
Where:
- \(\bar{x} = 62\) (sample mean)
- \(\mu_0 = 60\) (hypothesized population mean)
- \(s = 12\) (sample standard deviation)
- \(n = 16\) (sample size)

Substitute the given values:
\[
t = \frac{62 - 60}{\frac{12}{\sqrt{16}}} = \frac{2}{\frac{12}{4}} = \frac{2}{3} = 0.6667
\]

### Step 4: Find the degrees of freedom (df)

The degrees of freedom for a t-test is:
\[
df = n - 1 = 16 - 1 = 15
\]

### Step 5: Determine the critical t-value

Using a t-distribution table or a calculator for a two-tailed test with \(\alpha = 0.05\) and \(df = 15\), we find the critical t-values to be approximately \( \pm 2.131\).

### Step 6: Compare the test statistic with the critical t-value

- **Calculated t-statistic**: 0.6667
- **Critical t-values**: \(\pm 2.131\)

Since the calculated t-statistic (0.6667) lies within the range of \(-2.131 \leq t \leq 2.131\), we **fail to reject the null hypothesis**.

### Conclusion:
There is **insufficient evidence** to conclude that the true mean defect level is different from 60. Therefore, we fail to reject the null hypothesis at the 0.05 significance level.

Would you like further details or clarifications on this test?

### Related Questions:
1. How do you calculate the p-value for this hypothesis test?
2. What would happen if we used a one-tailed test instead?
3. How would the result change if the sample size were larger?
4. What assumptions are made when performing a t-test?
5. How do you interpret the confidence interval for the mean in this scenario?

**Tip**: When performing hypothesis testing, always ensure that the sample size is sufficiently large and that the sample comes from a normally distributed population or approximately normal, especially when using the t-test.

In a factory, the following null hypothesis is formulated for its defect levels. H0: mu<=60, H1: mu != 60. From an inspection report, the samples showed the following values, n = 16, sd = 12 and mean = 62. Test the hypothesis

Test the hypothesis that the true mean defect level is 60 using a two-tailed t-test. Given a sample size of 16, sample mean of 62, and sample standard deviation of 12, we calculate the t-statistic and compare it with critical values to draw conclusions. Suitable for Grades 11-12 students studying statistics.

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