A random sample of size n=16 is obtained from a normally...

A random sample of size $n=16$ is obtained from a normally distributed population with a population mean of $\mu=100$ and a variance of $\sigma^{2}=25$. a. What is the probability that $\bar{x}>101 ?$ b. What is the probability that the sample variance is greater than 45 ? c. What is the probability that the sample variance is greater than 60 ?

A random sample of size $n=16$ is obtained from a normally distributed population with a population mean of $\mu=100$ and a variance of $\sigma^{2}=25$.
a. What is the probability that $\bar{x}>101 ?$
b. What is the probability that the sample variance is greater than 45 ?
c. What is the probability that the sample variance is greater than 60 ?

Key Concepts

Chi-Square Distribution of the Sample Variance

When sampling from a normal population, the sample variance (when scaled appropriately) follows a chi-square distribution with degrees of freedom equal to one less than the sample size. Specifically, the statistic (n-1) times the sample variance divided by the population variance follows a chi-square distribution. This principle enables probability calculations regarding the sample variance and is crucial for making inferences about population variance in statistical analysis.

Sampling Distribution of the Sample Mean

When data are sampled from a normally distributed population with a known mean and variance, the sample mean follows a normal distribution. The mean of this sampling distribution is equal to the population mean, and its variance is the population variance divided by the sample size. This concept is essential for computing probabilities about the sample mean, such as the likelihood that it exceeds a certain value.

Z-Score and the Standard Normal Distribution

The Z-score transformation standardizes a normally distributed random variable. By converting the sample mean to a Z-score through subtracting the population mean and dividing by the standard error (the square root of the population variance divided by the sample size), one can use the standard normal distribution to find probabilities regarding the sample mean. This method is widely used in inferential statistics for hypothesis testing and confidence intervals.

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