A factory that manufactures bolts is performing a quality control experiment. Each object should have a length of no more than 20 centimeters. The factory believes that the length of the bolts exceeds this value and measures the length of 98  bolts. The sample mean bolt length was 20.03 centimeters. The population standard deviation is known to be 0.26  centimeters. what is the z- test statistic, p value and at alpha =.05 do we reject the hypthesis

This is a **one-sample z-test** for the population mean. The null hypothesis (\(H_0\)) and alternative hypothesis (\(H_1\)) can be stated as:

- \(H_0: \mu \leq 20\) (The true mean length of the bolts is 20 cm or less)
- \(H_1: \mu > 20\) (The true mean length of the bolts exceeds 20 cm)

### Given Information:
- Sample size (\(n\)) = 98
- Sample mean (\(\bar{x}\)) = 20.03 cm
- Population standard deviation (\(\sigma\)) = 0.26 cm
- Hypothesized population mean (\(\mu_0\)) = 20 cm
- Significance level (\(\alpha\)) = 0.05

### Step 1: Compute the z-test statistic

The formula for the z-test statistic is:

\[
z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]

Where:
- \(\bar{x}\) is the sample mean
- \(\mu_0\) is the hypothesized population mean
- \(\sigma\) is the population standard deviation
- \(n\) is the sample size

Let's calculate this now.The z-test statistic is approximately **1.14**.

### Step 2: Compute the p-value

Since this is a one-tailed test (\(H_1: \mu > 20\)), we need to find the p-value corresponding to this z-statistic using the standard normal distribution.

I'll calculate the p-value now.The p-value is approximately **0.127**.

### Step 3: Conclusion at \(\alpha = 0.05\)

- **Decision Rule**: If the p-value is less than \(\alpha = 0.05\), we reject the null hypothesis. Otherwise, we fail to reject it.
- In this case, the p-value (**0.127**) is greater than \(\alpha = 0.05\).

### Conclusion:
At the \(\alpha = 0.05\) significance level, we **fail to reject the null hypothesis**. This means there is not enough evidence to conclude that the true mean length of the bolts exceeds 20 cm.

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Would you like further details or have any questions?

Here are some related questions you might find helpful:
1. How is a z-test different from a t-test?
2. What assumptions must be met for using a z-test?
3. How can we interpret the p-value in hypothesis testing?
4. What happens if we use a two-tailed test instead of a one-tailed test?
5. How does increasing the sample size affect the z-test?

**Tip**: Always check if the population standard deviation is known. If not, use a t-test instead of a z-test for small sample sizes.

A factory that manufactures bolts is performing a quality control experiment. Each object should have a length of no more than 20 centimeters. The factory believes that the length of the bolts exceeds this value and measures the length of 98 bolts. The sample mean bolt length was 20.03 centimeters. The population standard deviation is known to be 0.26 centimeters. What is the z-test statistic, p-value, and at alpha = 0.05, do we reject the hypothesis?

Learn how to perform a one-sample z-test for bolt lengths in a quality control experiment. This problem involves hypothesis testing, calculating the z-test statistic, and interpreting the p-value to determine if the mean length of bolts exceeds 20 cm. Suitable for undergraduate statistics students.

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