It is known that the incomes of subscribers to a particular...

It is known that the incomes of subscribers to a particular magazine have a normal distribution with a standard deviation of $$\$ 6,600$$. A random sample of 25 subscribers is taken. a. What is the probability that the sample standard deviation of their incomes is more than $$\$ 4,000$$ ? b. What is the probability that the sample standard deviation of their incomes is less than $$\$ 8,000$$ ?

It is known that the incomes of subscribers to a particular magazine have a normal distribution with a standard deviation of $$\$ 6,600$$. A random sample of 25 subscribers is taken.
a. What is the probability that the sample standard deviation of their incomes is more than $$\$ 4,000$$ ?
b. What is the probability that the sample standard deviation of their incomes is less than $$\$ 8,000$$ ?

Key Concepts

Chi-square Distribution

The chi-square distribution is a continuous probability distribution that arises in statistics in the context of hypothesis testing and confidence interval estimation, particularly for variances. When data are normally distributed, the sum of the squared deviations from the mean follows a chi-square distribution with degrees of freedom equal to the number of independent pieces of information (typically the sample size minus one).

Sampling Distribution of the Variance

The sampling distribution of the variance describes the distribution of sample variance estimates computed from many random samples taken from a population. When the underlying data are normally distributed, the distribution of the sample variance, after being properly scaled by the population variance, follows a chi-square distribution. This concept is critical in inferential statistics for constructing tests and confidence intervals for the population variance.

Degrees of Freedom

Degrees of freedom refer to the number of independent values or quantities that can vary in an analysis without violating any given constraints. In the context of estimating the variance or standard deviation from a sample, the degrees of freedom are typically the sample size minus one, reflecting the fact that one parameter (the sample mean) is estimated from the data.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its mean and standard deviation, and it is fundamental in statistics due to the Central Limit Theorem, which explains why many independent random variables tend to be normally distributed when summed.

Sample Standard Deviation

The sample standard deviation is a measure of the dispersion or variability within a sample. It indicates how much the individual observations in a data sample deviate from the sample mean. Being an estimator of the population standard deviation, it is subject to sampling variability and has its own probability distribution when multiple samples are considered.

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