Question

Math

Posted 2 months ago

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Abdullah is a quality control expert at a factory that paints car parts. He knew that the factory was painting $20 \%$ of the parts with an error, so he made a change in the painting process. After the change, he wanted to test $H_{0}: p=0.2$ versus $H_{\mathrm{a}}: p<0.2$, where $p$ 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 $z=-1.25$.
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 ( $\hat{p}$ ) by dividing the number of parts with errors by the total number of parts sampled

step 2

$\hat{p} = \frac{70}{400} = 0.175$

step 3

Identify the z-score from the hypothesis test, which is given as $z = -1.25$

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 $z = -1.25$ 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 $\approx 0.1056$

step 7

Round the p-value to three decimal places as requested

step 8

P-value $\approx 0.106$

Answer

P-value $\approx 0.106$

Key Concept

P-value in Hypothesis Testing

Explanation

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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