A random sample of size n=18 is obtained from a normally...

A random sample of size $n=18$ is obtained from a normally distributed population with a population mean of $\mu=46$ and a variance of $\sigma^2=50$. a. What is the probability that the sample mean is greater than 50 ? b. What is the value of the sample variance such that $5 \%$ of the sample variances would be less than this value? c. What is the value of the sample variance such that $5 \%$ of the sample variances would be greater than this value?

A random sample of size $n=18$ is obtained from a normally distributed population with a population mean of $\mu=46$ and a variance of $\sigma^2=50$.
a. What is the probability that the sample mean is greater than 50 ?
b. What is the value of the sample variance such that $5 \%$ of the sample variances would be less than this value?
c. What is the value of the sample variance such that $5 \%$ of the sample variances would be greater than this value?

Key Concepts

Chi-Square Distribution

For samples drawn from a normal population, the sample variance (after being scaled by the degrees of freedom) follows a chi-square distribution. Specifically, multiplying the sample variance by (n ? 1) and dividing by the population variance yields a chi-square distributed variable with (n ? 1) degrees of freedom. This property is critical when determining the percentiles of the sample variance distribution.

Degrees of Freedom

Degrees of freedom refer to the number of independent pieces of information available to estimate another parameter, which in the context of the sample variance is equal to the sample size minus one. This concept is essential when relating the chi-square distribution to the sample variance, as the distribution's shape and critical values depend on the degrees of freedom.

Sampling Distribution of the Sample Mean

When samples are drawn from a normal population, the distribution of the sample mean is also normally distributed. Its mean is the same as the population mean, and its variance is equal to the population variance divided by the sample size. This concept allows one to assess the probability of the sample mean falling within any particular range by converting the sample mean to a standard normal variable.

Standardization

Standardization involves transforming a normally distributed variable into a standard normal variable by subtracting the mean and dividing by the standard deviation. This conversion enables the use of standard normal distribution tables or functions to calculate probabilities associated with the sample mean exceeding or falling below a given value.

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