In a large city it was found that summer electricity bills...

In a large city it was found that summer electricity bills for single-family homes followed a normal distribution with a standard deviation of $$\$ 100$$. A random sample of 25 bills was taken. a. Find the probability that the sample standard deviation is less than $$\$ 75$$. b. Find the probability that the sample standard deviation is more than $$\$ 150$$.

In a large city it was found that summer electricity bills for single-family homes followed a normal distribution with a standard deviation of $$\$ 100$$. A random sample of 25 bills was taken.
a. Find the probability that the sample standard deviation is less than $$\$ 75$$.
b. Find the probability that the sample standard deviation is more than $$\$ 150$$.

Key Concepts

Sampling Distribution of the Sample Variance

In a normally distributed population, the sample variance (or the square of the sample standard deviation) does not follow a normal distribution. Instead, when the population variance is known, the statistic formed by multiplying the sample variance by (n-1) and dividing by the true variance follows a chi-squared distribution. This result is foundational for making inferences about population variability using sample data.

Chi-Squared Distribution

The chi-squared distribution is a probability distribution that is widely used in inferential statistics, particularly for hypothesis testing and constructing confidence intervals for variance. When dealing with normally distributed data, the scaled sample variance follows a chi-squared distribution with degrees of freedom equal to the sample size minus one. It allows one to calculate the probabilities associated with observed values of sample variance or standard deviation.

Degrees of Freedom

Degrees of freedom refer to the number of independent values in a calculation that are free to vary. In the context of the sample variance calculated from a sample of size n, the degrees of freedom is n-1 because one parameter (usually the sample mean) is estimated from the data, reducing the number of independent pieces of information. This concept is crucial when using the chi-squared distribution for statistical inference.

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