Estimating a population mean (CI for 1 mean, sigma known)

The Confidence Interval for 1 Mean, σ Known:

What the different symbols mean:

[latex]n[/latex] is the sample size (number of people, items, etc… in the study)

[latex]\mu[/latex] is the population mean

[latex]\bar{x}[/latex] is the sample mean (also known as average)

[latex]\sigma[/latex] is the population standard deviation (sometimes we use this as an assumption)

[latex]E[/latex] is the margin of error

[latex]z_{\frac{\alpha}{2}}[/latex] is the critical value; we use the most common ones here based on the given confidence level

Assumptions when estimating 1 Mean, σ Known:

• We have a simple random sample
• We have a normal distribution OR [latex]n\ge 30[/latex]

Steps to construct the Confidence Interval for 1 Mean, σ Known:

• Identify all the symbols listed above (all the stuff that will go into the formulas). This includes [latex]n[/latex], [latex]\bar{x}[/latex], [latex]\sigma[/latex], the confidence level, and [latex]z_{\frac{\alpha}{2}}[/latex].
• Substitute values into the formula for [latex]E[/latex] and simplify to get the margin of error. Keep 3 or more decimal places for now (unless the number just cuts off on its own).
• Substitute values into the formula for the CI and simplify. Round your CI limits (at the end) to three significant digits (usually this means three decimal places). Sometimes the specific instructions for an exercise will tell you to round differently; if the instructions ask you to round shorter, that’s ok, but make sure you apply that AT THE END.

Example 1: Doordash

Suppose average DoorDash delivery times are normally distributed with a population standard deviation of σ = 6 minutes. A random sample of n = 40 restaurants is taken and has a sample mean delivery time of [latex]\bar{x}=42[/latex] minutes. Find a 90% confidence interval estimate for the population mean delivery time.

Solution

In this example, we are told that a random sample is part of the data. Additionally, we are told that the distribution is normal, AND we have a sample size of n = 40, so the requirements are met. Here we are given all the information we need. Let’s go through and identify the values we need to work this out.

• [latex]n = 40[/latex]
• [latex]\bar{x} = 42[/latex]
• [latex]\sigma = 6[/latex]
• [latex]z_{\frac{\alpha}{2}} = 1.645[/latex] since the confidence level is 90%
• [latex]E = z_{\frac{\alpha}{2}} \times\frac{\sigma}{\sqrt{n}} = 1.645 \times \frac{6}{\sqrt{40}} = 1.5606[/latex]

Here’s the confidence interval, in the three different forms:

Notice that in this example we did NOT convert into a percent. This problem deals with means (averages) which are not percents. So after we do the computation, we leave the numbers in decimal form. We just need to make sure we round according to instructions or by our default.

So what does this mean, what did we find out? We are 90% confident that the true population mean for DoorDash delivery times is between 40.439 minutes and 43.561 minutes.

Example 2: Red blood cell count

A simple random sample of n = 50 adults (including males and females) is obtained, and each person’s red blood cell count (in cells per microliter) is measured. The sample mean is [latex]\bar{x} = 4.63[/latex]. The population standard deviation for red blood cell counts is [latex]\sigma = 0.54[/latex]. Construct a 99% confidence interval estimate of the mean red blood cell count of adults.

Solution

In this example, we are told that a random sample is part of the data. Additionally, we are told that the sample size is n = 50, so the requirements are met. Here we are given all the information we need. Let’s go through and identify the values we need to work this out.

• [latex]n = 50[/latex]
• [latex]\bar{x} = 4.63[/latex]
• [latex]\sigma = 0.54[/latex]
• [latex]z_{\frac{\alpha}{2}} = 2.575[/latex] since the confidence level is 99%
• [latex]E = z_{\frac{\alpha}{2}} \times\frac{\sigma}{\sqrt{n}} = 2.575 \times \frac{0.54}{\sqrt{50}} = 0.1966[/latex]

Here’s the confidence interval, in the three different forms:

Notice that in this example we did NOT convert into a percent. This problem deals with means (averages) which are not percents. So after we do the computation, we leave the numbers in decimal form. We just need to make sure we round according to instructions or by our default.

So what does this mean, what did we find out? We are 99% confident that the true population mean for mean red blood cell count of adults is between 4.433 cells per microliter and 4.827 cells per microliter.

Example 3: youtubers^[1]

Imagine 35 randomly selected YouTubers have a mean salary of [latex]\bar{x} = \$60,000[/latex]. Also imagine the true standard deviation of their salaries is [latex]\sigma = \$19,000[/latex]. We are unsure that all YouTubers actually make $60,000. Use a 95% confidence level to construct a confidence interval estimate for the true mean yearly salary among YouTubers.

Solution

In this example, we are told that a random sample is part of the data. Additionally, we are told that the sample size is n = 35, so the requirements are met. Here we are given all the information we need. Let’s go through and identify the values we need to work this out.

• [latex]n = 35[/latex]
• [latex]\bar{x} = \$60,000[/latex]
• [latex]\sigma = \$19,000[/latex]
• [latex]z_{\frac{\alpha}{2}} = 1.96[/latex] since the confidence level is 95%
• [latex]E = z_{\frac{\alpha}{2}} \times\frac{\sigma}{\sqrt{n}} = 1.96 \times \frac{19000}{\sqrt{35}} = \$6,294.71[/latex]
• In this case we round to 2 places because US money has dollars and cents.

Here’s the confidence interval, in the three different forms:

Notice that in this example we did NOT convert into a percent. This problem deals with means (averages) which are not percents. So after we do the computation, we leave the numbers in decimal form. We just need to make sure we round according to instructions or by our default.

So what does this mean, what did we find out? We are 95% confident that the true mean yearly salary among  YouTubers is between $63,705.29 and $76,294.71.