Question

"I believe I have 21 of 30 widgets that are defective. I look at 9 of these widgets and I find that 4 of those widgets are defective, and 5 are not. Should I reject my initial believe. I need to set this up as a hypothesis test, and then accept or reject the null according to the choice of my alpha."

Where do I start with this? I think I understand the question but just want to make sure... Lets say I chose a two tailed test with an alpha of 5%?  

Answer 1

Ho:p=21/30=0.70

Ha p not =0.7

two tail

alpha=0.05

z=p^-p/sqrt(p*(1-p)/n

p^=sample proportion=x/n=4/9=0.4444444

z=(0.4444444-0.70)/sqrt(0.70*(1-0.70)/9)

z=-1.673004

test statistic=-1.673004

p=2*(=NORM.S.DIST(-1.673004;TRUE))=2*0.047163254=

p= 0.09432651

p>0.05

Fail to reject Ho.

Accept Ho.

There is sufficient evidence at 5% to support the belief that  21 of 30 widgets that are defective

Answer 2

To set up a hypothesis test for this scenario, you would start by defining the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis represents your initial belief, and the alternative hypothesis challenges that belief.

Let's define the hypotheses:

Null Hypothesis (H0): The proportion of defective widgets is equal to or less than the initial belief of 21/30. Ha: The proportion of defective widgets is greater than the initial belief of 21/30.

Next, you need to choose a test statistic and a significance level (alpha) for your hypothesis test. In this case, since you are dealing with proportions, you can use the z-test for proportions.

Since you want to perform a two-tailed test with an alpha of 5%, you need to divide the alpha by 2 to get the significance level for each tail:

Significance level (alpha) = 0.05 (for each tail)

Now, you need to calculate the test statistic based on the sample data to make the decision.

1. Calculate the sample proportion of defective widgets from the 9 widgets you examined: Sample proportion (p̂) = Number of defective widgets / Total number of widgets examined Sample proportion (p̂) = 4/9 ≈ 0.4444

2. Calculate the standard error of the sample proportion: Standard Error (SE) = √(p̂(1 - p̂) / n) SE = √(0.4444 * (1 - 0.4444) / 9) ≈ 0.1652

3. Calculate the z-score for the sample proportion: z = (Sample proportion - Population proportion under H0) / Standard error z = (0.4444 - 21/30) / 0.1652 ≈ -1.900

4. Look up the critical z-value for a one-tailed test with alpha = 0.05 (or 5%): The critical z-value is approximately -1.645 (you can find this value using a standard normal table or calculator).

5. Make the decision: Since the calculated z-score (-1.900) is less than the critical z-value (-1.645), we reject the null hypothesis.

Conclusion: Based on the sample data and the chosen significance level, we have enough evidence to reject the initial belief that the proportion of defective widgets is 21/30. The data suggests that the proportion of defective widgets is likely greater than 21/30.