A simple random sample of 50 items from a population with σ=6 resulted in a sample mean of 32 .

step 1 All right, this question gives us sample size, standard deviation, sample mean, and it wants us

A simple random sample of 50 items from a population with...

A simple random sample of 50 items from a population with $\sigma=6$ resulted in a sample mean of $32 .$ a. Provide a 90$\%$ confidence interval for the population mean. b. Provide a 95$\%$ confidence interval for the population mean. c. Provide a 99$\%$ confidence interval for the population mean.

A simple random sample of 50 items from a population with $\sigma=6$ resulted in a sample mean of $32 .$
a. Provide a 90$\%$ confidence interval for the population mean.
b. Provide a 95$\%$ confidence interval for the population mean.
c. Provide a 99$\%$ confidence interval for the population mean.

Key Concepts

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean represents the probability distribution of all possible sample means of a given size drawn from the population. According to the Central Limit Theorem, this distribution will be approximately normal if the sample size is large, even if the underlying population distribution is not normal, which justifies the use of the Z-distribution in constructing confidence intervals.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. It provides an estimated range that is likely to include the true population mean with a specified probability, such as 90%, 95%, or 99% confidence levels.

Standard Error

The standard error is a measure of the amount of variability or dispersion in the sampling distribution of a statistic, most commonly the sample mean. It is computed as the population standard deviation divided by the square root of the sample size, indicating how much the sample mean is expected to vary from the true population mean.

Z-Distribution and Critical Value

When the population standard deviation is known and the sample size is sufficiently large, the Z-distribution is used to construct confidence intervals. The critical value from the standard normal distribution corresponds to the desired level of confidence and determines the width of the confidence interval by scaling the standard error.

Transcript

Please give Ace some feedback