4.1 Statistical Inference and Confidence Intervals

Sampling Distribution for the Mean

A researcher takes repeated samples of size 1,000 from the residents of New York to collect data on mean income of residents of New York.

For each sample of size 1,000, we can calculate a sample mean, $\bar{x}$. If the researcher were to take 50 such samples (each of sample size 1,000), a series of sample means can be calculated:

A probability distribution based on all possible random samples of a certain size from a population—or sampling (or sample) distribution—can then be analyzed. For example, we can calculate the mean of these sample means, and we can calculate the standard deviation of these sample means.

There are two important properties of the sample distribution of these sample means:

1. The mean of the sample means (notated as ${\mu }_{\bar{x}}$) is equal to the population mean $\mu$.
   1. Written mathematically: ${\mu }_{\bar{x}}=\mu$
2. The standard deviation of the sample means (notated as ${\sigma }_{\bar{x}}$) is equal to the population standard deviation $\sigma$ divided by the square root of the sample size $n$.
   1. Written mathematically: ${\sigma }_{\bar{x}}=\frac{\sigma }{\sqrt{n}}$

The central limit theorem describes the relationship between the sample distribution of sample means and the underlying population. This theorem is an important tool that allows data scientists and researchers to use sample data to generate inferences for population parameters.

Conditions of the central limit theorem:

• If random samples are taken from any population with mean $\mu$ and standard deviation, where the sample size is at least 30, then the distribution of the sample means approximates a normal (bell-shaped) distribution.
• If random samples are taken from a population that is normally distributed with mean $\mu$ and standard deviation $\sigma$, then the distribution of the sample means approximates a normal (bell-shaped) distribution for any sample size

Confidence Interval for the Mean When the Population Standard Deviation Is Known

Although in many situations the population standard deviation is unknown, in some cases a reasonable estimate for the population standard deviation can be obtained from past studies or historical data.

Here is what is needed to calculate the confidence interval for the mean when the population standard deviation is known:

1. A random sample is selected from the population.
2. The sample size is at least 30, or the underlying population is known to follow a normal distribution.
3. The population standard deviation ($\sigma$) is known.

Once these conditions are met, the margin of error $E$ is calculated according to the following formula:

Where:

$z_c$ is called the critical value of the normal distribution and is calculated as the z-score, which includes the area corresponding to the confidence level between $-z_c$ and $+z_c$. For example, for a 95% confidence interval, the corresponding critical value is $z_c=1.96$ since an area of 0.95 under the normal curve is contained between the z-scores of $-1.96$ and $+1.96$.

$\sigma$ is the population standard deviation.

$n$ is the sample size.

Note: $\frac{\sigma }{\sqrt{n}}$ is called the standard error of the mean.

Typical values of $z_c$ for various confidence levels are shown in Table 4.2.

Graphically, the critical values can be marked on the normal distribution curve, as shown in Figure 4.2. (See Discrete and Continuous Probability Distributions for a review of the normal distribution curve.) Figure 4.2 is an example for a 95% confidence interval where the area of 0.95 is centered as the area under the standard normal curve showing the values of $-z_c$ and $+z_c$. Since the standard normal curve is symmetric, and since the total area under the curve is known to be 1, the area in each of the two tails can be calculated to be 0.025. Thus, the critical value is that z-score which cuts off an area of 0.025 in the upper tail of the standard normal curve.

Figure 4.2 Critical Values of z_c Marked on Standard Normal Curve for 95% Confidence Interval

Once the margin of error has been calculated, the confidence interval can be calculated as follows:

$\text{Sample Mean}\pm \text{Margin of Error}$

Note that this is the general format for constructing any confidence interval, namely the margin of error is added and subtracted to the sample statistic to generate the upper and lower bounds of the confidence interval, respectively. A sample statistic describes some aspect of the sample, such as a sample mean or sample proportion.

General Form of a Confidence Interval: $\text{Sample Statistic}\pm \text{Margin of Error}$

Confidence Interval for the Mean When the Population Standard Deviation Is Unknown

A confidence interval can still be determined when the population standard deviation is unknown by calculating a sample standard deviation based on the sample data. This is actually the more common application of confidence intervals.

Here are the conditions required to use this procedure:

1. A random sample is selected from the population.
2. The sample size is at least 30, or the underlying population is known to follow a normal distribution.
3. The population standard deviation ($\sigma$) is unknown; the sample standard deviation $\left(s\right)$ can be calculated.

Once these requirements are met, the margin of error (E) is calculated according to the following formula:

$t_c$ is called the critical value of the t-distribution. The t-distribution is a bell-shaped, symmetric distribution similar to the normal distribution, though the t-distribution has “thicker tails” as compared to the normal distribution. The comparison of the normal and t-distribution curves is shown in Figure 4.3.

Figure 4.3 Comparison of t-Distribution and Normal Distribution Curves

The t-distribution is actually a family of curves, determined by a parameter called degrees of freedom (df), where df is equal to $n-1$. The critical value $t_c$ is similar to a z-score and specifies the area under the t-distribution curve corresponding to the confidence level between $-t_c$ and $+t_c$. Values of $t_c$ can be obtained using either a look-up table or technology.

For example, for a 95% confidence interval and sample size of 30, the corresponding critical value is $t_c=\mathrm{2.045}$ since an area of 0.95 under the t-distribution curve is contained between the t-scores of $-2.045$ and $+2.045$.

$s$ is the sample standard deviation.

$n$ is the sample size.

df is degrees of freedom, where $df=n-1$.

Typical values of $t_c$ for various confidence levels and degrees of freedom are shown in Table 4.3.

Note that Python can be used to calculate these critical values. Python provides a function called t.ppf() that generates the value of the t-distribution corresponding to a specified area under the t-distribution curve and specified degrees of freedom. This function is part of the scipy.stats library.

The syntax for the function is:
t.ppf(area to left, degrees of freedom)

For example, for a 95% confidence interval, there is an area of 0.95 centered under the t-distribution curve, which leaves a remaining area of 0.05 for the two tails of the distribution, which implies an area of 0.025 in each tail. To find a critical value corresponding to the upper 0.025 area, note that the area to the left of this critical value will then be 0.975.

Here is an example of Python code to generate the critical value $t_c$ for a 95% confidence interval and 15 degrees of freedom.

    # import function from scipy.stats library
    from scipy.stats import t
    
    # define parameters called area to left and degrees of freedom df
    Area_to_left = 0.975
    df = 15
    
    # use t.ppf function to calculate critical values associated with t-
    # use round function to round answer to 3 decimal places
    round (t.ppf(Area_to_left, df), 3)
    

2.131

Once the margin of error is calculated, the confidence interval is formed in the same way as the previous section, namely:

Confidence Interval for Proportions

We can also calculate a confidence interval for a population proportion based on the use of sample data. The basis for the confidence interval will be the application of a normal approximation to the binomial distribution. Recall from Discrete and Continuous Probability Distributions that a binomial distribution is a probability distribution for a discrete random variable where there are only two possible outcomes of an experiment. A proportion measures the number of successes in a sample. For example, if 10 survey respondents out of 50 indicate they are planning to travel internationally within the next 12 months, then the proportion is 10 out of 50, which is 0.2, or 20%. Note that the term success does not necessarily imply a positive outcome. For example, a researcher might be interested in the proportion of smokers among U.S. adults, and the number of smokers would be considered the number of successes.

Some terminology will be helpful:
$p$ represents the population proportion, which is typically unknown.
$\hat{p}$ represents the sample proportion.
$x$ represents the number of successes in the sample.
$n$ represents the sample size.

Here are the requirements to use this procedure:

1. A random sample is selected from the population.
2. Verify that the normal approximation to the binomial distribution is appropriate by ensuring that both $n\hat{p}$ and $n(1-\hat{p})$ are both at least 5, where $\hat{p}$ represents the sample proportion.

The sample proportion $\hat{p}$ is calculated as the number of successes divided by the sample size:

When these requirements are met, the margin of error (E) for a confidence interval for proportions is calculated according to the following formula:

Where:

$z_c$ is called the critical value of the normal distribution and is calculated as the z-score, which includes the area corresponding to the confidence level between $-z_c$ and $+z_c$.

$\hat{p}$ represents the sample proportion.

$n$ is the sample size.

Once the margin of error is calculated, the confidence interval is formed in the same way as the previous section. In this case, the point estimate is the sample proportion $\hat{p}$:

Sample Size for Confidence Interval for the Mean

When determining a confidence interval for a mean, a researcher can use the margin of error formula for the mean; solving this formula algebraically for the sample size ($n$), the following minimum sample size formula can be used to determine the minimum sample size needed to achieve a certain margin of error:

Sample size formula for confidence interval for mean:

Where:
$z_c$ is the critical value of the normal distribution.
$\sigma$ is the population standard deviation.
$E$ is the desired margin of error.

Note that for sample size calculations, sample size results are rounded up to the next higher whole number. For example, if the formula previously shown results in a sample size of 59.2, then the sample size will be rounded to 60.

Sample Size for Confidence interval for a Proportion

In Example 4.5, how would the researcher know the optimal sample size to use to collect sample data on the proportion of adults who are smokers? The margin of error formula can be used to derive the sample size for a proportion as follows:

Where:
$z_c$ is the critical value of the normal distribution.
$\hat{p}$ is the sample proportion.
$E$ is the desired margin of error.

Notice in this formula, it’s assumed that some prior estimate for the sample proportion $\hat{p}$ is available, perhaps from historical data or previous studies. If a prior estimate for the sample proportion is not available, then use the value of 0.5 for $\hat{p}$.