A sample of 64 measurements is taken from a continuous...

A sample of 64 measurements is taken from a continuous population, and the sample mean is found to be 32.0. The standard deviation of the population is known to be $2.4 .$ An interval estimation is to be made of the mean with a level of confidence of $90 \%$. State or calculate the following items. a. $\bar{x}$ b. $\sigma$ c. $n$ d. $1-\alpha$ e. $z(\alpha / 2)$ f. $ \cdot \sigma_{x}$ g. $E$ (maximum error of estimate) h. Upper confidence limit i. Lower confidence limit

A sample of 64 measurements is taken from a continuous population, and the sample mean is found to be 32.0. The standard deviation of the population is known to be $2.4 .$ An interval estimation is to be made of the mean with a level of confidence of $90 \%$. State or calculate the following items.
a. $\bar{x}$
b. $\sigma$
c. $n$
d. $1-\alpha$
e. $z(\alpha / 2)$
f. $ \cdot \sigma_{x}$
g. $E$ (maximum error of estimate)
h. Upper confidence limit
i. Lower confidence limit

Key Concepts

Margin of Error

The margin of error represents the maximum expected difference between the true population mean and the sample mean. It is computed by multiplying the critical value by the standard error.

Standard Error

The standard error is an estimate of the standard deviation of the sampling distribution of the sample mean. It is calculated by dividing the population standard deviation by the square root of the sample size.

Critical Value (z-score)

The critical value is derived from the standard normal distribution and corresponds to the chosen significance level divided by two (?/2). This value determines the number of standard errors to extend on either side of the sample mean to form the confidence interval.

Confidence Interval

The confidence interval is a range constructed around the sample mean, extending an equal distance in both directions by the margin of error. It is the interval within which the true population mean is expected to lie with a predefined level of confidence.

Sample Size

The sample size is the number of observations drawn from the population. A larger sample size decreases the standard error and leads to a more precise estimate of the population mean.

Population Standard Deviation

This is the known measure of variability in the entire population from which the sample is drawn. Its known value simplifies interval estimation since the sampling distribution of the mean will follow a normal distribution.

Sample Mean

The sample mean is the arithmetic average of the observed data points in the sample. It serves as a point estimate of the population mean and is used as the center point when constructing the confidence interval.

Confidence Level

The confidence level represents the proportion of intervals that would contain the true population mean if the sampling process were repeated numerous times. It is expressed as 1 - ?, where ? is the significance level.

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