Paired Means

One-sample mean refers to a statistical analysis that involves comparing the mean of a single sample to a known or hypothesized population mean. In other words, it is a statistical technique used to determine whether the mean of a sample is significantly different from a known or hypothesized value.

This analysis is typically carried out using a t-test, which involves calculating a test statistic (t-value) based on the difference between the sample mean and the population mean, as well as the standard error of the mean. The t-value is then compared to a critical value from a t-distribution, and if it exceeds this value, it suggests that the sample mean is significantly different from the population mean.

The Central Limit Theorem states that if the sample size is large (typically, n > 30), then the distribution of sample means will be approximately normally distributed with a standard deviation (i.e., standard error) equal to the population standard deviation divided by the square root of the sample size (SE = σ / √n). In other words, as the sample size increases, the standard error decreases, which means that the sample mean becomes a more precise estimate of the population mean.

When constructing confidence intervals and conducting hypothesis tests we will usually be using the t distribution when working with one mean. The only exception would be in cases where σ is known. This scenario is most common in the fields of education and psychology where some tests are normed to have a certain μ and σ. In those cases, the z distribution can be used.

In terms of language, all of these tests could be called “single sample mean tests” or “one sample mean tests.” We could also specify the sampling distribution by using the term “single sample mean t test” or “single sample mean z test.”

The flow chart below may help you in determining which method should be used when constructing a sampling distribution for one sample mean.

One Sample Mean

Identify when z and t distributions should be used.

• Is the population known to be normally distributed?
• Is the population standard deviation known?
• Is the sample size at least 30?

 

t Distribution

The t-distribution is a type of probability distribution that arises when estimating the mean of a normally distributed population with an unknown standard deviation. It is similar in shape to the normal distribution but has fatter tails, which means that it has more probability in the tails than the normal distribution.

The t-distribution is used in situations where the population standard deviation is unknown and must be estimated from the sample. It is also used when the sample size is small (less than 30), and the assumption of normality is not met.

In mathematical notation, the t-distribution with degrees of freedom df is written as t(df). The degrees of freedom refer to the number of independent observations in the sample, and determine the shape and scale of the t-distribution. As the degrees of freedom increase, the t-distribution approaches the normal distribution.

The below plot compares the standard normal distribution (i.e., z distribution) to a t distribution. The solid blue line is the standard normal distribution and the dashed red line is a t distribution with 2 degrees of freedom. Here, the tails of the t distribution are higher than the tails of the normal distribution.

This plot compares the standard normal distribution to a t distribution with 10 degrees of freedom. Notice that the two distributions are becoming more similar as the sample size increases.

 

The next plot compares the standard normal distribution to a t distribution with 30 degrees of freedom.

In the final graph, the standard normal distribution is compared to a t distribution with 500 degrees of freedom. Here, the two distributions are nearly identical. As the degrees of freedom approach infinity, the t distribution approaches (i.e., becomes more similar to) the standard normal distribution.

 

Conduct inference for one sample mean

One sample mean is a statistical test used to determine whether a sample mean is statistically different from a hypothesized population mean. This type of hypothesis test is commonly used in research studies to test a specific hypothesis about a population based on a single sample.

Confidence Intervals

Confidence intervals are used to estimate unknown population parameters. Because the population standard deviation (α) will almost always be unknown in situations in which we are constructing confidence intervals for means, the t-distribution is used to estimate the sampling distribution. The following pages will show you how to construct a confidence interval for a population mean using formulas. Similar to how we computed necessary minimum sample sizes for confidence intervals for proportions, we will also compute the necessary minimum sample size for constructing a confidence interval for a mean.

This interval is calculated using the sample data and is commonly used in research studies to estimate the population mean based on a single sample.

Let’s review some of symbols and equations that we learned in previous tutorials:

The formula for calculating a confidence interval for one sample mean is:

sample mean ± (critical value x standard error)

where the critical value is determined based on the desired level of confidence and the standard error is calculated as:

CI = x̄ ± t-value * (s / sqrt(n))

where x̄ is the sample mean, s is the sample standard deviation, n is the sample size, t-value is the appropriate value from the t-distribution for the desired confidence level and degrees of freedom, and CI represents the confidence interval for the population mean.

To calculate a confidence interval for one sample mean, the following steps can be followed:

1. Determine the sample size (n), sample mean (x̄), and sample standard deviation (s).

2. Select the confidence level. A commonly used confidence level is 95%, which corresponds to alpha (α) = 0.05.

3. Determine the critical value (t) from the t-distribution with n-1 degrees of freedom and alpha/2 as the probability in each tail. For example, for a 95% confidence interval and a sample size of n=30, the critical value is 2.045.

4. Calculate the margin of error using the formula: Margin of error = t * (s / sqrt(n))

5. Calculate the lower and upper bounds of the confidence interval using the formulas:

Lower bound = x̄ – margin of error

Upper bound = x̄ + margin of error

The resulting interval provides an estimate of the population mean with a certain level of confidence. For example, a 95% confidence interval for a sample mean of 50 with a sample size of 30 and a sample standard deviation of 10 would be:

Lower bound = 50 – (2.045 * (10 / sqrt(30))) = 44.94
Upper bound = 50 + (2.045 * (10 / sqrt(30))) = 55.06

Therefore, we can be 95% confident that the true population mean falls between 44.94 and 55.06.

Hypothesis Testing

The hypothesis test is used to determine whether there is sufficient evidence to reject or fail to reject the null hypothesis. The null hypothesis states that there is no significant difference between the sample mean and the population mean. The alternative hypothesis states that there is a significant difference between the sample mean and the population mean.

Here we will be using the five step hypothesis testing procedure to compare the mean in one random sample to a specified population mean using the normal approximation method.

1. Check assumptions and write hypotheses

Data must be quantitative. In order to use the t distribution to approximate the sampling distribution either the sample size must be large (n≥30) or the population must be known to be normally distributed. The possible combinations of null and alternative hypotheses are:

2. Calculate the test statistic

For the test of one group mean we will be using a t test statistic:

t = (x̄ – μ0) / (s / sqrt(n))

where x̄ is the sample mean, μ0 is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.

3. Determine the p-value

When testing hypotheses about a mean or mean difference, a t distribution is used to find the p-value. These t distributions are indexed by a quantity called degrees of freedom, calculated as df = n-1 for the situation involving a test of one mean or test of mean difference.

4. Make a decision

• If p ≤ α, reject the null hypothesis (there is evidence to support the alternative hypothesis).
• If p > α, fail to reject the null hypothesis (there is not enough evidence to support the alternative hypothesis).

If significance level α is not mentioned then consider α = 0.05

Rejection Region: If the p-value is less than the level of significance (alpha), then the null hypothesis is rejected. The rejection region is defined by the critical values from the standard normal distribution.

Acceptance Region: If the p-value is greater than the level of significance (alpha), then the null hypothesis is not rejected. The acceptance region is defined by the complement of the rejection region.

5. State a “real world” conclusion

Based on all the 4 steps above, we should write a sentence or two concerning our decision in relation to the original research question.