Construct the appropriate confidence interval. A simple...

Construct the appropriate confidence interval. A simple random sample of size $n=12$ is drawn from a population that is normally distributed. The sample mean is found to be $\bar{x}=45$ and the sample standard deviation is found to be $s=14 .$ Construct a $90 \%$ confidence interval about the population mean.

Construct the appropriate confidence interval.
A simple random sample of size $n=12$ is drawn from a population that is normally distributed. The sample mean is found to be $\bar{x}=45$ and the sample standard deviation is found to be $s=14 .$ Construct a $90 \%$ confidence interval about the population mean.

Key Concepts

Standard Error

The standard error is a measure of the variability of the sample mean as an estimate of the population mean. It is calculated by dividing the sample standard deviation by the square root of the sample size. The magnitude of the standard error influences the width of the confidence interval.

Degrees of Freedom

Degrees of freedom refer to the number of independent values in a calculation that are free to vary. In the context of estimating a population mean using a t-distribution, the degrees of freedom are typically calculated as the sample size minus one, which impacts the critical value used in constructing the confidence interval.

Margin of Error

The margin of error represents the extent of the interval on either side of the sample mean within which the true population mean is expected to lie with a certain level of confidence. It is determined by multiplying the critical value from the t-distribution by the standard error.

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true value of an unknown population parameter. It provides a quantifiable measure of the uncertainty associated with a sample estimate, and the level of confidence reflects the probability that the interval computed from repeated sampling would capture the true parameter.

t-Distribution

The t-distribution is used when making inferences about the population mean when the sample size is small and the population standard deviation is unknown. It accounts for increased uncertainty in the estimate of the standard deviation, and approaches the normal distribution as the sample size increases.

Transcript

00:01 The following is a solution to number seven, and this says that a random sample of 12 people were selected.

00:05 We don't really know what it is, but it's from a normal approximation or a normal distribution, approximately normal.

00:11 So that allows us to have this small sample size.

00:13 Normally you have to have a sample size of at least 30, but since it comes from a normal distribution, we can just say that this is okay.

00:19 So n equals 12 is fine.

00:21 And the sample mean x bar was 45, and the sample standard deviation is 14.

00:27 And we're asked to find the 90 % confidence interval for the population mean.

00:31 So since we're estimating the population mean, we're either going to use the z or the t interval.

00:35 And we're going to use the t interval because we don't know what sigma is.

00:39 We don't know that population standard deviation because we weren't given that.

00:43 We were only given the sample standard deviation s.

00:46 So because of that, we have to use the t interval.

00:50 Okay, so t interval.

00:52 So the 90 % confidence interval.

00:55 Now you can use the t interval formula.

00:57 You can certainly go about it that way.

00:59 I'm going to use the calculator just because it goes by a little bit more quickly...

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