8.1 A Single Population Mean (Known σ)

Parameter values, such as population mean or population proportion, are generally unknown. For example, if we wanted to know the percentage of college students who skip breakfast in the morning, then to know this actual percentage we would have to survey all college students (also need to define who qualifies as a college student). This is not quite easily attainable. The best we can do is to take a representative sample from the population of college students and use the percentage of students in the sample who skip breakfast as an estimate for the entire college student population. However, since a sample doesn’t include every member of the population, these sample statistics are often going to be slightly off from the actual population value of the parameter. The difference between the sample statistic and the population parameter is called the sampling error. We expect the sample results to naturally vary from one sample to another, but knowing the sampling distribution of the sample statistics from the Central Limit Theorem (CLT) allows us to calculate the maximum value of the sample error for different levels of desired confidence. This maximum error of estimate is called the Margin of Error (ME). The margin of error depends on the standard error of the sampling distribution of the sample statistic estimating the parameter and the desired level of confidence:

Margin of Error = (Critical Value) • (Standard Error)

Standard Errors (SE)

Recall that the CLT says the sampling distribution of sample statistics will be normal or approximately normal if CLT requirements are met.

For means, use the Standard Error of the mean = the standard deviation of the sampling distribution of sample means ([latex]\bar x[/latex])

Standard Error of the mean [latex]= \sigma_{\bar x} = \dfrac{\sigma}{\sqrt n}[/latex].

If the population standard deviation,[latex]\sigma[/latex], is unknown, we can use the sample standard deviation, [latex]s[/latex], as a substitute for [latex]\sigma[/latex]. This substitution works well for larger sample sizes. We’ll discuss the approach for smaller sample sizes in the next section.

Estimating an Unknown Parameter

In general, we can estimate parameters in a population which are generally unknown with their corresponding sample statistics. A sample statistic used to estimate a population parameter is called a point estimate. We can represent a point estimate as a point on a number line.

When estimating an unknown parameter using a sample statistic, there’s always going to be uncertainty involved due to variability in sampling. What are the chances that our sample statistic exactly equals what we’re trying to estimate in the population? Very little. So, instead we build an interval around the point estimate (sample statistic) by going a certain distance (called Margin of Error) to its left and the same distance to the right. We use that interval to estimate the unknown parameter. This interval is called a confidence interval.

Computing Confidence Intervals
We compute confidence intervals as follows:

• Start with your point estimate: [latex]\bar x[/latex] (x-bar) or [latex]\hat p[/latex] (p-hat).
• Subtract the Margin of Error (ME) from the point estimate to obtain the Left endpoint of the confidence interval
• Add the Margin of Error (ME) to the point estimate to obtain the right endpoint of the confidence interval
  Note that the Margin of Error (ME) is also known as EBM (for means) and as EBP (for proportions)

Three Ways to Write a Confidence Interval
Once you compute the interval, you can represent the interval in multiple ways.
Three Ways to Write a Confidence Interval  © mathbootcamps.com (Link Reported Not working). Try this archive link instead.

1. Plus minus notation
2. Interval Notation
3. Inequality/tri-linear Notation

If you need algebra refresher on how to write interval notations: Interval Notation @Coolmath

Changing the Confidence Level or Sample Size

Effect of Changing the Confidence Level (Example 8.4)

• Increasing the confidence level increases the error bound, making the confidence interval wider.
• Decreasing the confidence level decreases the error bound, making the confidence interval narrower.

Effect of Changing the Sample Size (Example 8.5)

• Increasing the sample size causes the error bound to decrease, making the confidence interval narrower.
• Decreasing the sample size causes the error bound to increase, making the confidence interval wider.

Sample Size Calculations

According to Central Limit Theorem, sample means follow a normal distribution with a mean equal to the population mean and a standard error of \[\sigma_{\bar x} = \frac{\sigma}{\sqrt n}\] Recall that margin of error is given by \[ME = z_c\cdot (\text{Standard Error})\] that is \[ME = z_c\cdot\frac{\sigma}{\sqrt n} = \frac{z_c\cdot\sigma}{\sqrt n}\]

We solve this for [latex]n[/latex]. Start by multiplying both sides by [latex]\sqrt n[/latex], and then divide the resulting equation by ME on both sides. \[\sqrt n = \frac{z_c\cdot\sigma}{ME}\] Squaring both sides results in \[n = \left(\frac{z_c\cdot\sigma}{ME}\right)^2\]
Calculating the required sample size using this formula will almost always result in a decimal value for [latex]n[/latex]. Since the sample size has to be a positive integer, we will need to do some rounding. We will always round up the final answer for sample size calculations^[1] and indicate this in the formula by using the ceiling function (round up to the nearest integer):
\[n = \left\lceil\left(\frac{z_c\cdot\sigma}{ME}\right)^2\right\rceil\]