Construct the appropriate confidence interval. A simple...

Construct the appropriate confidence interval. A simple random sample of size $n=40$ is drawn from a population. The sample mean is found to be $\bar{x}=120.5$ and the sample standard deviation is found to be $s=12.9 .$ Construct a $99 \%$ confidence interval about the population mean.

Construct the appropriate confidence interval.
A simple random sample of size $n=40$ is drawn from a population. The sample mean is found to be $\bar{x}=120.5$ and the sample standard deviation is found to be $s=12.9 .$ Construct a $99 \%$ confidence interval about the population mean.

Key Concepts

Degrees of Freedom

Degrees of freedom refer to the number of independent values that can vary in estimating a statistical parameter. In the context of estimating a mean with an unknown standard deviation, it is typically calculated as the sample size minus one, which adjusts for the loss of one degree of freedom when estimating the sample standard deviation.

Confidence Interval

A confidence interval provides a range of plausible values for a population parameter, such as the mean, based on sample data. It is constructed so that under repeated sampling, a certain percentage (e.g., 99%) of these intervals will contain the true parameter, representing the uncertainty in the estimation process.

t-Distribution

When the population standard deviation is unknown and only the sample standard deviation is available, the t-distribution is used to model the sampling distribution of the sample mean. This distribution accounts for extra variability due to estimating the standard deviation from the sample and is characterized by degrees of freedom, usually equal to the sample size minus one.

Margin of Error

The margin of error quantifies the extent of uncertainty around the sample estimate. It is computed by multiplying the appropriate critical value from the t-distribution by the standard error of the mean. This value is then added to and subtracted from the sample mean to form the confidence interval.

Sampling Distribution of the Mean

The sampling distribution of the mean is the distribution formed by calculating the means of many samples drawn from the same population. It plays a crucial role in inferential statistics, enabling the use of the Central Limit Theorem to justify the approximation of the sampling distribution to a normal or t-distribution, depending on sample size and known parameters.

Transcript

00:01 The following is a solution in number 9, and this says that we have a random sample of 40.

00:06 Now, it doesn't say the population distribution, but it doesn't matter because that sample size is big enough to assume normality and assume the central limit theorem.

00:14 So had this been a sample of like size 20 or 25, then we would actually have to say, but it comes from a normal population.

00:23 But since that sample size is 40, it doesn't matter how skewed or whatever the population is, 40 is big.

00:30 Enough.

00:30 And the mean of that sample is 120 .5 and the standard deviation of that sample is 12 .9.

00:37 And we're asked to find the 99 % confidence interval for the population mean mu.

00:42 So first off, let's decide what method to use.

00:44 We're going to use the t interval here.

00:49 So the t interval.

00:50 The reason why we use a t interval is because we're estimating the population mean.

00:53 So it's either going to be the z or the t interval.

00:55 And the reason why we can't use the z interval is because we don't know what sigma is.

00:59 We don't know that population standard deviation.

01:02 We only know the sample standard deviation s.

01:04 So whenever you don't know that sigma, you don't know that population standard deviation, you have to use the t interval...

Please give Ace some feedback