Question

What three assumptions must be met when you are using the \(z\) test to test differences between two means when \(\sigma_{1}\) and \(\sigma_{2}\) are known?

Short answer

The three assumptions are normality of distribution, known population variances, and independent samples.

Step-by-step solution

Understanding the Assumptions

Before using the \(z\) test to test the difference between two means, it is crucial to be aware of the necessary assumptions. These assumptions ensure the validity of the results derived from the \(z\) test. Each assumption addresses a particular aspect of the data distribution or sampling process.

Assumption 1 - Normality of Distribution

The first assumption is that the data from each group should be drawn from a normally distributed population. This means that the distribution of the data around the mean for both groups should follow the normal distribution (bell-shaped curve), especially when dealing with small sample sizes. However, with a large sample size, the Central Limit Theorem implies normality.

Assumption 2 - Known Population Variances

The second assumption is that the population variances, \(\sigma_1^2\) and \(\sigma_2^2\), are known. This is essential for calculating the standard error of the difference in means accurately. When variances are unknown and need estimation from the sample, a different test, such as the \(t\) test, may be more appropriate.

Assumption 3 - Independent Samples

The third assumption is that the samples from the two populations are independent. This means that the data collected from one group should not influence or be related to the data from the other group. Independence ensures that the sample values are unbiased and the \(z\) test can validly assess the difference between groups.

Key concepts

Normal Distribution

When using a \(z\) test, one pivotal assumption is that the data collected comes from a normally distributed population. In simpler terms, this means that if you were to plot a graph based on the data, it would resemble a bell-shaped curve. This shape is indicative of a normal distribution. Why does this matter? Well, in statistics, the normal distribution provides a foundation for making predictions about data trends.

Here are some key points about normal distribution:

• Data points tend to be symmetrically distributed around the mean.
• Most values lie close to the mean, with fewer situated at either extreme end of the scale.

In practice, even if the population itself isn't perfectly normal, the Central Limit Theorem comes to the rescue. It indicates that for large sample sizes, the distribution of the sample mean will resemble a normal distribution, even if the population distribution is not perfectly normal. This is especially reassuring when working with diverse data sets.

Population Variances

Another cornerstone of the \(z\) test is the necessity to know the population variances, often represented as \(\sigma_1^2\) and \(\sigma_2^2\). Knowing these variances allows for an accurate calculation of the standard error of the difference between two means.

But what exactly are population variances?

• They measure how much the data in each population varies from the mean.
• Smaller variances indicate that the data points are clustered closely around the mean, while larger variances imply more spread out data.

Why is this assumption crucial for a \(z\) test? Without knowing population variances, the reliability of the results may be compromised. When variances are unknown, it is advisable to consider alternative methods like the \(t\) test, which estimates variance from the sample data.

Independent Samples

The assumption of independent samples is a vital requirement for the \(z\) test. It states that the samples drawn from two different populations must be independent of one another. Simply put, the outcome or results from one sample should not affect or be related to the results from the other sample.

Here's why it matters:

• Independent samples ensure unbiased results, permitting valid comparative analysis.
• If samples are dependent, any inference made about the population parameters may be flawed.

This concept becomes almost second nature in statistics. You always want to ensure that your comparison between groups stands on solid ground, and independence of samples enables that.