Question

A company makes car batteries and claims 80% of its ABC batteries are good for 70 months or longer. Assume that this claim is true. Let p ˆ be the proportion in a sample of 100 such ABC batteries. What is the probability that this sample proportion is within 0.05 of the population proportion.

Answer 1

Answer:

78.88% probability that this sample proportion is within 0.05 of the population proportion

Step-by-step explanation:

We need to understand the normal probability distribution and the central limit theorem to solve this question.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For proportion p in a sample of size n, we have that $\mu = p, s = \sqrt{\frac{\pi(1-\pi)}{n}}$

In this question:

$p = 0.8, n = 100$

So

$\mu = 0.8, s = \sqrt{\frac{0.8*0.2}{100}} = 0.04$

What is the probability that this sample proportion is within 0.05 of the population proportion.

This is the pvalue of Z when X = 0.8 + 0.05 = 0.85 subtracted by the pvalue of Z when X = 0.8 - 0.05 = 0.75.

X = 0.85

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{0.85 - 0.8}{0.04}$

$Z = 1.25$

$Z = 1.25$ has a pvalue of 0.8944.

X = 0.75

$Z = \frac{X - \mu}{s}$

$Z = \frac{0.75 - 0.8}{0.04}$

$Z = -1.25$

$Z = -1.25$ has a pvalue of 0.1056.

0.8944 - 0.1056 = 0.7888

78.88% probability that this sample proportion is within 0.05 of the population proportion