Z Test

• A z-test is a statistical method used to test whether the averages of two groups differ, provided the data follow a normal distribution. 
   
• To perform a z-test, first set up the null and alternative hypotheses. Next, calculate the value of the z-test statistic. The decision about whether to accept or reject the null hypothesis is made by comparing the test statistic to the z critical value.
   
• A z-test is performed on a population whose data follow a normal distribution, and the sample size is at least 30 with independent observations. This test is used to determine whether the means of two populations are equal when the population variances are known. 
   
• The null hypothesis can be rejected if the calculated z-test statistic exceeds the critical value, indicating a meaningful difference between the population means.

Key Takeaways from Z-Tests

Z tests are utilized in comparing one sample to another or with a population in various sectors. Let’s look at a few key takeaways:

• Z- tests are applied when the sample sizes are more than 30.
   
• These tests are used only when the standard deviation is known, otherwise we prefer t-tests.
   
• Using these, we compare averages and proportions in data sets.
   
• To draw conclusions from data, we assess assumptions using the null hypothesis (H₀) and the alternative hypothesis (H₁).

Z-tests come in various types. We will now explore each of them:

One-sample z-test: We use the one-sample test when a single sample differs extensively from a known population mean. To use this test, the population standard deviation should be known, and the sample size must be more than 30.

Two-sample z-test: This test compares the means of two independent samples.

Let μ₁ and μ₂ be the population means; X₁ and X₂ are sample mean.

The null hypothesis of the test could be:

H₀: μ₁ – μ₂ = 0

H₁: μ₁ – μ₂ ≠ 0

We calculate the z-test score using the formula:

\(Z = \frac{(\bar{X}_1 - \bar{X}_2) - (\mu_1 - \mu_2)} {\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\)

Here:

X = sample mean

σ₁ and σ₂ = standard deviation

n₁ and n₂ = population sample sizes for p₁ and p₂ 

Z Test Formula

The z-test formula compares the z-statistic with the z-critical value to determine if there is a difference between the means of two populations. In hypothesis testing, the z-critical value separates the distribution into acceptance and rejection regions. If the test statistic falls in the rejection region, the null hypothesis is rejected; otherwise, it is not rejected.

The formulas used to set up hypothesis tests for both one-sample and two-sample z tests are given as:



One-Sample Z Test

A one-sample z test is used to determine whether the sample mean differs significantly from the population mean when the population standard deviation is known.

The formula for calculating the z-test statistic is:

\(z = \frac{\overline{x} - \mu}{\frac{\sigma}{\sqrt{n}}}\)

Where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

The algorithm to set a one-sample z test based on the z test statistic can be given as:

Left-Tailed Test:

Null Hypothesis: \(H_0 : μ =μ_0\)
Alternate Hypothesis: \(H_1:μ<μ_0\)
The decision criterion is to reject the null hypothesis if the z statistic is less than the z critical value.

Right-Tailed Test:

Null Hypothesis: \(H_0: μ = μ_0\)
Alternate Hypothesis: \(H_1: μ > μ_0\)
The decision criterion is to reject the null hypothesis if the z statistic is greater than the z critical value.

Two-Tailed Test:

Null Hypothesis: H0 : μ = μ0
Alternate Hypothesis: \(H_1 : μ ≠ μ_0\)
The decision criterion is to reject the null hypothesis if the z statistic is greater than the z critical value.

Two-Sample Z Test

A two-sample z test is a statistical method used to determine whether there is a significant difference between the means of two independent samples. It checks whether the averages of two groups differ, assuming the population variances are known and the samples are drawn from normally distributed populations.

The z test statistic formula is expressed as:

\(z = \frac{\left( \bar{x}_1 - \bar{x}_2 \right) - \left( \mu_1 - \mu_2 \right)} {\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \)

Where \(\bar{x}_1, μ_1, \sigma_1^2\) are the sample mean, population mean and population variance respectively for the first sample. Similarly, \(\bar{x}_2, μ_2, \sigma_2^2\) are the sample mean, population mean, and population variance of the second sample.

The two-sample z test is similar to the one-sample test but is used to compare the means of two different samples. For instance, the null hypothesis can be stated as, \(H_0: μ_1 = μ_2\), meaning the means of the two populations are equal.

Z tests can be used in comparing proportions between a sample and a population, or two samples. For example: Determining whether the proportion of girls in a school differs significantly from the percentage worldwide.

Z Test Proportions:

Z tests can be used to compare proportions between a sample and a population, or between two samples. For example, determining whether the proportion of girls in a school differs significantly from the percentage worldwide.

One Proportion Z Test:
A one-proportion z test is applied when comparing an observed proportion with a theoretical proportion across two groups.
The z test statistic for this test is calculated using the following formula:

\(z = \frac{p - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \)

where,
p is the observed proportion, 
\(P_0\) is the theoretical (or hypothesized) proportion, and 
N is the sample size.

The null hypothesis states that the two proportions are equal, while the alternative hypothesis states that they differ.

Two Proportion Z Test:

A two-proportion z-test compares two proportions to determine whether they differ significantly.

The formula for the z test statistic is:

\(z = \frac{p_1 - p_2 - 0}{\sqrt{p(1 - p)\left( \frac{1}{n_1} + \frac{1}{n_2} \right)}} \)

Where, \(p = \frac{x_1 + x_2}{n_1 + n_2} \)

Where \(P_1\) and \(P_2\) are the sample proportions for the two groups, \(n_1\) and \(n_2\) are the sample sizes, and P is the pooled proportion calculated from both groups combined.

How to Calculate Z Test Statistic?

Interpreting the problem correctly is the most crucial step in calculating the z-test statistic. It is essential to identify the appropriate tailed test and determine which category the z statistic falls into before proceeding.

The steps to calculate the z-test statistic are given as;

Step 1: Identify the type of test. For example, if comparing the means of two populations in one direction, a right-tailed two-sample z test is used.

Step 2: Set up the hypotheses:

\(H_0: \mu_1 = \mu_2, H_1: \mu_1 > \mu_2.\)

Step 3: Using the z table, find the critical value for the given alpha level, e.g., 1.645.

Step 4: Calculate the z-test statistic with the formula:

\(z = \frac{ \left( \bar{x}_1 - \bar{x}_2 \right) - \left( \mu_1 - \mu_2 \right) }{ \sqrt{ \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} } } \)

Substitute the values:

\(X_1 =22.1\), \(\sigma _1 =4.8\), \(n_1 =60\), \(X_2 = 18.8\), \(\sigma _2 =8.1\), \(n_2 =40\) and \(\mu_1-\mu_2 = 0.\) thus, \(z = 2.32.\)

The calculated z statistic is 2.32.

Step 5: Compare the calculated z statistic with the critical value. Since 2.32 > 1.645, reject the null hypothesis.

Step 6: Conclude that there is sufficient evidence to support the claim that the students' scores in the first class are higher.

The comparison between z-test and t-test is given as;
 

Learning about the Z-test helps students compare datasets in real-world situations. The tips and tricks mentioned below will help us understand the concept easily:
 

• The z-tests are used only when the sample size is greater than 30.
   
• Correct formulas should be used for z-tests and t-tests.
   
• Use the decision rule to make quick decisions: 
   
• If \(∣Z∣ < Z_{critical},\) reject H_0​ (significant difference).
   
• If \(∣Z∣ < Z_{critical},\) fail to reject H₀​ (no significant difference).
   
• There is a trick to determine the normality: If \(n < 30\), check for normality before doing a z-test.

Z tests are used in various fields in comparing averages. Let’s explore a few of them:
 

• Z-tests can be used to compare the average scores of students in a school with those in another school to analyze if they are significantly different.
   
• Companies apply z-tests to analyze and compare the effectiveness of two advertisements.
   
• Researchers can use these tests to determine which drug or medicine yields a better result.
   
• These are widely used by teachers to check the effectiveness of different teaching methods.
   
• Banks can make use of these tests in analyzing the default loan rates of two specific loans.
   
• In medical research, Z-tests are used to compare the effects of a treatment between two groups.