A random sample of 225 measurements is selected from a...

A random sample of 225 measurements is selected from a population, and the sample mean and standard deviation are $\bar{x}=32.5$ and $s=30.0$, respectively. a. Use a $99 \%$ confidence interval to estimate the mean of the population, $\mu$. b. How large a sample would be needed to estimate $m$ to within 5 with $99 \%$ confidence? *c. Use a $99 \%$ confidence interval to estimate the population variance, $\sigma^2$. d. What is meant by the phrase $99 \%$ confidence as it is used in this exercise?

A random sample of 225 measurements is selected from a population, and the sample mean and standard deviation are $\bar{x}=32.5$ and $s=30.0$, respectively.
a. Use a $99 \%$ confidence interval to estimate the mean of the population, $\mu$.
b. How large a sample would be needed to estimate $m$ to within 5 with $99 \%$ confidence?
*c. Use a $99 \%$ confidence interval to estimate the population variance, $\sigma^2$.
d. What is meant by the phrase $99 \%$ confidence as it is used in this exercise?

Key Concepts

Interpretation of the Confidence Level

The confidence level of an interval estimation, such as 99%, refers to the proportion of times that the calculated confidence interval would contain the true population parameter if the same procedure were repeated over numerous samples. It is not a probability statement about a single interval or the parameter itself, but rather a measure of the reliability of the estimation method across repeated sampling.

Confidence Interval for a Population Variance

Unlike the estimation of a mean, constructing a confidence interval for a population variance involves the chi-square distribution. This approach provides a range within which the true population variance is expected to lie with a specified confidence level, making it a key concept in understanding the dispersion or spread of data in a population. It demonstrates the application of different probability distributions in interval estimation depending on the parameter being estimated.

Confidence Interval for a Population Mean

A confidence interval is a range of values, derived from a sample, that is likely to contain the true value of an unknown population parameter. In the context of estimating a population mean, this interval is constructed by taking the sample mean and adding and subtracting a margin of error, which is determined by the desired level of confidence and the variability in the data. This concept is central to inferential statistics, allowing conclusions about the population based on sample data.

Margin of Error and Sample Size Determination

The margin of error quantifies the maximum expected difference between the true population parameter and a sample estimate. When planning studies, sample size is calculated based on a desired margin of error and level of confidence, ensuring the accuracy and precision of the estimate. This process typically involves using critical values from a statistical distribution and an estimate of population variability, thus linking the margin of error directly to the required sample size.

Transcript

00:01 So we have a sample of 400 elements, and we know that the mean of those values comes out to be 40 .5 with a sample standard deviation of 26.

00:10 And so in part a, we want a 99 % confidence interval for the mean.

00:17 And so because this is a large sample, we'll say that this value is approximately equal to the standard deviation.

00:24 So we can take our 40 .5 plus or minus our 0 .0 .0 .0 .5.

00:30 Value for 99 % confidence, which means we'll have 0 .005 in the upper tail, and that's 2 .576 times our sample standard deviation divided by the square root of n.

00:42 And when we do that calculation, we get the interval to be, and we get the interval to be 37 .15 to 43 .85.

00:59 Now on part b, we want to find what size n do we need to have that sampling error, that plus or minus part, be equal to 0 .5 with 99 % confidence.

01:14 So we know that that will end up being, we know that this value, if we take the z value times that sample standard deviation, which we're going to assume again is that over the square root of n, and that's our sampling error.

01:28 When we do our solving for n, we get n is equal to the z squared times the sigma squared divided by the sampling error squared.

01:37 So we'll still use that 2 .576, and we'll use that 26 as our estimate and our sampling error of 0 .5.

01:48 And so in effect, all those values are squared.

01:52 So i can type in 2 .576 times 26 divided by 0 .5 .6.

01:59 And then just square that whole quantity after i've gotten that product in quotient.

02:04 And we need a pretty big sample size.

02:07 We need to round that up 17 ,943 .13 .13...