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Posted 2 months ago

Abdullah is a quality control expert at a factory that paints car parts. He knew that the factory was painting 20%20 \% of the parts with an error, so he made a change in the painting process. After the change, he wanted to test H0:p=0.2H_{0}: p=0.2 versus Ha:p<0.2H_{\mathrm{a}}: p<0.2, where pp is the proportion of parts that the factory was painting with an error.

Abdullah took a sample of 400 parts and found that 70 had an error, and the corresponding test statistic was z=1.25z=-1.25.

Assuming that the necessary conditions are met, what is the approximate P-value for Abdullah's significance test?
You may round to three decimal places.
P-value \approx
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Answer from Sia
Posted 2 months ago
Solution by Steps
step 1
Calculate the sample proportion ( p^\hat{p} ) by dividing the number of parts with errors by the total number of parts sampled
step 2
p^=70400=0.175\hat{p} = \frac{70}{400} = 0.175
step 3
Identify the z-score from the hypothesis test, which is given as z=1.25z = -1.25
step 4
Use the standard normal distribution table or a calculator to find the p-value corresponding to the z-score
step 5
The p-value is the area to the left of z=1.25z = -1.25 in the standard normal distribution
step 6
Looking up the p-value in the standard normal distribution table or using a calculator, we find that the p-value 0.1056\approx 0.1056
step 7
Round the p-value to three decimal places as requested
step 8
P-value 0.106\approx 0.106
P-value 0.106\approx 0.106
Key Concept
P-value in Hypothesis Testing
The p-value represents the probability of obtaining a test statistic as extreme as the one observed, under the assumption that the null hypothesis is true. In this case, it is the probability of observing a z-score of -1.25 or lower if the true proportion of errors is 0.2.

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