Hypothesis Testing in Manufacturing: Analyzing Daily Output

QUESTION

1. A manufacturer thinks her mean daily output is 650 units. A sample of 81 days reveals a mean of 632 units with delta square = 5184. Testing the hypothesis at alpha=.10 which of the following statement are true.

 (1) The manufacturer should reject the null hypothesis.

 (2) To test hypothesis, the correct z test statistic for comparison is 1.96, 

(3) the standard error is 5.76 

(4) The manufacturer’s statement is correct and the output is statistically the same as 650 at alpha=.10

ANSWER

 Hypothesis Testing in Manufacturing: Analyzing Daily Output

Introduction

Hypothesis testing is a crucial statistical method used by manufacturers to make informed decisions about their production processes. In this essay, we will explore the hypothesis testing process applied to a manufacturer who believes her mean daily output is 650 units. A sample of 81 days has been taken, and based on this sample, we will determine whether the manufacturer’s claim is supported or rejected. We will use a significance level of alpha = 0.10 to test the hypothesis.

Hypothesis Statement

The null hypothesis, denoted as H0, states that the mean daily output is equal to 650 units. The alternative hypothesis, denoted as Ha, opposes the null hypothesis, suggesting that the mean daily output is different from 650 units.

 The manufacturer should reject the null hypothesis

To determine whether the null hypothesis should be rejected, we will conduct a hypothesis test. The sample mean provided is 632 units, and the sample size is 81 days. The population standard deviation, denoted as σ (sigma), can be estimated using the formula:

σ ≈ √(delta square / n)

where delta square is the given value of 5184 (sample variance), and n is the sample size of 81.

σ ≈ √(5184 / 81) ≈ √64 ≈ 8

Next, we will calculate the standard error (SE) of the sample mean using the formula:

SE = σ / √n

SE ≈ 8 / √81 ≈ 8 / 9 ≈ 0.889

Now, we can calculate the z-test statistic using the formula:

z = (sample mean – population mean) / SE

z = (632 – 650) / 0.889 ≈ -20 / 0.889 ≈ -22.5

Since we are testing at a significance level of alpha = 0.10, we look for the critical z-value corresponding to this level. A two-tailed test at alpha = 0.10 has critical z-values of approximately ±1.645. Since the calculated z-value (-22.5) falls beyond the critical values, we can reject the null hypothesis.

 To test the hypothesis, the correct z-test statistic for comparison is 1.96

This statement is not true. The critical z-value for a two-tailed test at alpha = 0.10 is approximately ±1.645, not 1.96. The z-test statistic value we calculated earlier (-22.5) is far beyond the critical region, reinforcing our decision to reject the null hypothesis.

 The standard error is 5.76

This statement is not true. The correct calculation of the standard error is approximately 0.889, as calculated earlier. A standard error of 5.76 is not applicable in this scenario.

The manufacturer’s statement is correct, and the output is statistically the same as 650 at alpha = 0.10:
This statement is false. Based on the hypothesis test, we have enough evidence to reject the manufacturer’s claim that the mean daily output is 650 units. The data suggests that the true mean is significantly different from 650 units, as the z-test statistic (-22.5) falls well beyond the critical region at alpha = 0.10.

Conclusion

In conclusion, the manufacturer’s belief that her mean daily output is 650 units is not supported by the data. The hypothesis test conducted at alpha = 0.10 led us to reject the null hypothesis, indicating that the true mean daily output is statistically different from 650 units. It is important for manufacturers to use hypothesis testing and statistical analysis to make informed decisions about their production processes, ensuring optimal efficiency and quality.