Question

General Co. has designed a new tire, and they don’t know what the average amount of tread life is going to be. The tread life is normally distributed with a standard deviation of 295.5 kilometers.
i) If the company samples 800 tires and records their tread life, what is the probability the sample mean is between the true mean and 350 kilometers over the true mean?
ii) How large a sample must be taken to be 95 percent sure the sample mean will be within 150 kilometers of the true mean?

Answer 1

Answer on Question #45577 – Math – Statistics and Probability

Problem.

General Co. has designed a new tire, and they don't know what the average amount of tread life is going to be. The tread life is normally distributed with a standard deviation of 295.5 kilometers.

i) If the company samples 800 tires and records their tread life, what is the probability the sample mean is between the true mean and 350 kilometers over the true mean?

ii) How large a sample must be taken to be 95 percent sure the sample mean will be within 150 kilometers of the true mean?

Solution.

Suppose that tread life is normally distributed with mean $\mu$ and standard deviation $\sigma = 295.5$ ($N(\mu, \sigma)$).

i) The sample of $n = 800$ tires (random variable $X$) has distribution $N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)$.

The probability that the sample mean is between the true mean and 350 kilometers over the true mean is

$$P(\mu < X < \mu + 350).$$

The corresponding transformation formula is $Z = \frac{X - \mu}{\sigma / \sqrt{n}}$, where $Z$ is distributed normally, $Z \sim N(0,1)$.

$$P(\mu < X < \mu + 350) = P\left(0 < Z < \frac{350}{295.5 \cdot \sqrt{800}}\right) = P(0 < Z < 0.041) \approx 0.0164.$$

ii) $z^*$ for 95% confidence level equals $z^* = 1.96$. Then the confidence interval is

$$\left(\mu - z^* \frac{\sigma}{\sqrt{n}}, \mu + z^* \frac{\sigma}{\sqrt{n}}\right)$$

Then the sample mean will be within 150 kilometers of the true mean if $z^* \frac{\sigma}{\sqrt{n}} < 150$ or

$$n > \left(\frac{1.96 \cdot 295.5}{150}\right)^2 \approx 14.9. \text{ Hence } n \geq 15.$$

Answer: i) 0.0164. ii) $n \geq 15$.

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