Types of Z-Test: One-Sample and Two-Sample Z-Test

In statistical analysis, the Z-test is a widely used hypothesis test that helps determine whether there is a significant difference between sample data and the population data. It is particularly useful when the sample size is large and the population standard deviation is known. In this blog, we will discuss two common types of Z-tests: One-Sample Z-Test and Two-Sample Z-Test. We will also cover the formulas for both tests along with examples to help you understand their applications.

What is a Z-Test?

A Z-test is a statistical test that uses the Z-distribution (a normal distribution) to determine whether there is a significant difference between a sample mean and a population mean, or between the means of two samples. It is typically used when:

• The sample size is large (n > 30).
• The population standard deviation is known.
• The data is approximately normally distributed.

There are two primary types of Z-tests:

1. One-Sample Z-Test
2. Two-Sample Z-Test

1. One-Sample Z-Test

A One-Sample Z-Test is used when you are comparing the sample mean to a known population mean. This test helps determine if there is enough statistical evidence to support the claim that the sample mean differs from the population mean.

Formula for One-Sample Z-Test

The formula to calculate the Z-statistic for a one-sample Z-test is:

Z = (x̄ - μ) / (σ /sqrt(n))

Where:

• ( x̄ ) = Sample mean
• ( μ ) = Population mean (under the null hypothesis)
• ( σ ) = Population standard deviation
• ( n ) = Sample size

Steps for Performing a One-Sample Z-Test:

1. State the Hypotheses:

   • Null Hypothesis (H₀): ( mu = μ ) (The population mean is equal to the hypothesized value).
   • Alternative Hypothesis (H₁): ( mu ≠ μ ) (The population mean is not equal to the hypothesized value).

2. Set the Significance Level (( alpha )): Commonly set at 0.05.

3. Calculate the Z-Statistic using the formula.

4. Compare the Z-Statistic to the critical Z-value (obtained from Z-tables based on ( alpha )).

5. Make a Decision: If the absolute value of the Z-statistic is greater than the critical value, reject the null hypothesis.

Example of a One-Sample Z-Test:

Suppose a manufacturer claims that the average weight of their product is 500 grams. You collect a random sample of 100 products and find that the sample mean weight is 505 grams, with a population standard deviation of 15 grams. You want to test at a 0.05 significance level whether the sample mean is significantly different from the claimed mean of 500 grams.

• Sample mean (( x̄ )) = 505 grams
• Population mean (( μ )) = 500 grams
• Population standard deviation (( σ )) = 15 grams
• Sample size (( n )) = 100
• Significance level (( alpha )) = 0.05

Step-by-Step Calculation:

1. State the Hypotheses:

   • H₀: ( mu = 500 ) grams
   • H₁: ( mu ≠ 500 ) grams (Two-tailed test)

2. Calculate the Z-Statistic:

   Z = (505 - 500) / 15 / sqrt(100) = 3.33

1. Determine the Critical Z-Value:

   • For a two-tailed test at ( alpha = 0.05 ), the critical Z-value is ( pm 1.96 ) (obtained from standard Z-tables).

2. Decision:

   • Since the calculated Z-value (3.33) is greater than 1.96, we reject the null hypothesis.

3. Conclusion:

   • There is enough evidence to conclude that the mean weight of the product is significantly different from the claimed 500 grams.

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2. Two-Sample Z-Test

A Two-Sample Z-Test is used to compare the means of two independent samples to determine if there is a significant difference between them. This test is often used when you want to compare two groups (e.g., control group vs. experimental group) or two different treatments.

Formula for Two-Sample Z-Test

The formula for the two-sample Z-test is:

Z = (x̄_1 - x̄_2) / sqrt (σ_1^2 /n_1 + σ_2^2 /n_2)

Where:

• ( x̄_1 ) = Sample mean of group 1
• ( x̄_2 ) = Sample mean of group 2
• ( σ_1 ) = Population standard deviation of group 1
• ( σ_2 ) = Population standard deviation of group 2
• ( n_1 ) = Sample size of group 1
• ( n_2 ) = Sample size of group 2

Steps for Performing a Two-Sample Z-Test:

1. State the Hypotheses:

   • Null Hypothesis (H₀): ( mu_1 = mu_2 ) (The means of both groups are equal).
   • Alternative Hypothesis (H₁): ( mu_1 ≠ mu_2 ) (The means of both groups are different).

2. Set the Significance Level (( alpha )): Typically set at 0.05.

3. Calculate the Z-Statistic using the formula.

4. Compare the Z-Statistic to the critical Z-value (using standard Z-tables).

5. Make a Decision: If the absolute value of the Z-statistic exceeds the critical value, reject the null hypothesis.

Example of a Two-Sample Z-Test:

Suppose a researcher wants to compare the test scores of two different teaching methods. The following data is collected:

• Group 1 (Method A): Sample mean (( x̄_1 )) = 78, Population standard deviation (( σ_1 )) = 10, Sample size (( n_1 )) = 40
• Group 2 (Method B): Sample mean (( x̄_2 )) = 74, Population standard deviation (( σ_2 )) = 12, Sample size (( n_2 )) = 50
• Significance level (( alpha )) = 0.05

Step-by-Step Calculation:

1. State the Hypotheses:

   • H₀: ( mu_1 = mu_2 ) (The mean test scores of both groups are the same)
   • H₁: ( mu_1 ≠ mu_2 ) (The mean test scores of both groups are different)

2. Calculate the Z-Statistic:

   Z = (78 - 74) / sqrt (10^2 /40 + 12^2 / 50) = 1.72

1. Determine the Critical Z-Value:

   • For a two-tailed test at ( alpha = 0.05 ), the critical Z-value is ( pm 1.96 ).

2. Decision:

   • Since the calculated Z-value (1.72) is less than the critical value (1.96), we fail to reject the null hypothesis.

3. Conclusion:

   • There is not enough evidence to conclude that the teaching methods have different effects on the test scores.

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Conclusion

In summary, both the One-Sample Z-Test and the Two-Sample Z-Test are used to compare sample data to a known population or between two sample groups. The One-Sample Z-Test is used when comparing the sample mean to a known population mean, while the Two-Sample Z-Test is used when comparing the means of two independent samples.

By understanding the formulas and the step-by-step procedures for these tests, you can confidently conduct hypothesis tests and draw meaningful conclusions from your data.

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Happy testing!