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

Q: 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 $\approx$

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the observed sample proportion ($\hat{p}$) and the null hypothesis proportion ($p_0$). ‖ ‖ step 2 ⋮ Calculate the observed sample proportion: $\hat{p} = \frac{\text{number of errors}}{\text{sample size}} = \frac{70}{400} = 0.175$. ‖ ‖ step 3 ⋮ The null hypothesis states that $p_0 = 0.2$. ‖ ‖ step 4 ⋮ The test statistic given is $z = -1.25$. ‖ ‖ step 5 ⋮ Use the standard normal distribution to find the P-value corresponding to $z = -1.25$. ‖ ‖ step 6 ⋮ The P-value is the area to the left of $z = -1.25$ in the standard normal distribution. ‖ ‖ step 7 ⋮ Using the standard normal distribution table or a calculator, find the area to the left of $z = -1.25$. ‖ ‖ step 8 ⋮ The approximate P-value is $P(Z < -1.25) \approx 0.1056$. ‖ ∻Answer∻ ⚹ P-value $\approx$ 0.106 ⚹ ∻Key Concept∻ ⚹P-value in Hypothesis Testing⚹ ∻Explanation∻ ⚹The P-value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct. In this case, it represents the probability of observing a sample proportion of errors as low as 0.175 or lower, given that the true proportion is 0.2.⚹◊What is the formula for calculating the margin of error in a one-sample z interval for a proportion?⍭ Generate me a similar question◊

Question

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 $\approx$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the observed sample proportion ($\hat{p}$) and the null hypothesis proportion ($p_0$). ‖ ‖ step 2 ⋮ Calculate the observed sample proportion: $\hat{p} = \frac{ ext{number of errors}}{ ext{sample size}} = \frac{70}{400} = 0.175$. ‖ ‖ step 3 ⋮ The null hypothesis states that $p_0 = 0.2$. ‖ ‖ step 4 ⋮ The test statistic given is $z = -1.25$. ‖ ‖ step 5 ⋮ Use the standard normal distribution to find the P-value corresponding to $z = -1.25$. ‖ ‖ step 6 ⋮ The P-value is the area to the left of $z = -1.25$ in the standard normal distribution. ‖ ‖ step 7 ⋮ Using the standard normal distribution table or a calculator, find the area to the left of $z = -1.25$. ‖ ‖ step 8 ⋮ The approximate P-value is $P(Z < -1.25) \approx 0.1056$. ‖ ∻Answer∻ ⚹ P-value $\approx$ 0.106 ⚹ ∻Key Concept∻ ⚹P-value in Hypothesis Testing⚹ ∻Explanation∻ ⚹The P-value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct. In this case, it represents the probability of observing a sample proportion of errors as low as 0.175 or lower, given that the true proportion is 0.2.⚹◊What is the formula for calculating the margin of error in a one-sample z interval for a proportion?⍭ Generate me a similar question◊

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