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 ? 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 $n=16$ 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 ?
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

Quantile Calculation and Probability Thresholds

Quantiles in probability distributions, such as those of the chi-square distribution, are used to determine threshold values below or above which a certain percentage of the distribution falls. This method is instrumental in establishing confidence bounds and understanding the variability of sample statistics.

Chi-Square Distribution of the Sample Variance

For a normally distributed population, it is known that the statistic formed by multiplying the sample variance by (n-1) and dividing by the population variance follows a chi-square distribution with degrees of freedom equal to n-1. This property is crucial for making inferences about the population variance based on the sample variance.

Standard Error and Z-Score Transformation

Standard error is the measure of the variability of the sample mean from the population mean, computed as the population standard deviation divided by the square root of the sample size. The z-score transformation standardizes the difference between the sample mean and the population mean, allowing the use of the standard normal distribution to find probabilities.

Sampling Distribution of the Sample Mean

This concept involves understanding that when a sample is drawn from a normally distributed population, the distribution of the sample mean is normal with the same mean as the population and a standard error equal to the population standard deviation divided by the square root of the sample size. This forms the basis for calculating probabilities regarding the sample mean using z-scores.

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