"The charge life of a certain lithium ion battery for camcorders is normally distributed, with mean 90 minutes and standard deviation 35 minutes.


b. Describe the sampling distribution of x̄, the sample mean charge life for a random sample of 10 such batteries."

Verified step by step guidance

1. Identify the population distribution: The charge life of the lithium ion battery is normally distributed with mean \( \mu = 90 \) minutes and standard deviation \( \sigma = 35 \) minutes.
2. Recognize that the sampling distribution of the sample mean \( \overline{x} \) for a sample size \( n = 10 \) will also be normally distributed because the original population is normal.
3. Calculate the mean of the sampling distribution of \( \overline{x} \), which is the same as the population mean: \( \mu_{\overline{x}} = \mu = 90 \).
4. Calculate the standard deviation of the sampling distribution of \( \overline{x} \), called the standard error (SE), using the formula: \n\n\( \sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}} = \frac{35}{\sqrt{10}} \).
5. Summarize the sampling distribution: \( \overline{x} \) is normally distributed with mean \( 90 \) and standard deviation \( \frac{35}{\sqrt{10}} \).

Key Concepts

Normal Distribution

A normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve defined by its mean and standard deviation. It describes how values of a variable are distributed, with most values clustering around the mean. In this question, the battery life follows a normal distribution, which allows for specific probability calculations.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean is the probability distribution of the means of all possible samples of a given size from a population. It describes how the sample mean varies from sample to sample. For a normal population, the sampling distribution of the sample mean is also normal, with the same mean as the population but a smaller standard deviation.

Standard Error of the Mean

The standard error of the mean measures the variability of the sample mean from the population mean and is calculated as the population standard deviation divided by the square root of the sample size. It quantifies the precision of the sample mean as an estimate of the population mean. In this problem, it helps describe the spread of the sampling distribution for the sample mean of 10 batteries.