Question

# On the basis of past history, a car manufacturer knows that 10% (p = .10) of all newly made cars have an initial defect. In a random sample of n = 100 recently made cars, 13%$(\hat{p} = .13)$have defects. Find the value of the standardized statistic (z-score) for this sample proportion.

Solution

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We will use the formula for the $z$-score for a sample proportion:

$z=\frac{\hat p - p}{\sqrt{\frac{p\cdot(1-p)}{n}}} ,$

where $p$ is the population proportion, $n$ is the sample size and $\hat p$ is the value of sample statistic (proportion).

Inserting the values $p=0.1$ ($10\%$ of $\textbf{all}$ newly made cars have an initial defect, so this is the population parameter), $n=100$ (a sample of 100 newly made cars was taken), and $\hat p=0.13$ ($13\%$ of cars $\textbf{from the sample}$ had defects) into the formula above yields the value of $z$-score for a sample proportion:

$\boxed{z}=\frac{0.13-0.1}{\sqrt{\frac{0.1\cdot0.9}{100}}}=\boxed{1} .$

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