What is a Z-Test in Mathematics?
A Z-Test is a statistical test used to determine whether there is a significant difference between sample and population means when the population variance is known and the sample size is large. It is often used in hypothesis testing to ascertain if observed data deviates significantly from what is expected.
When Should a Z-Test be Used?
A Z-Test is appropriate to use under the following conditions:- The sample size is large (n > 30).- The population standard deviation is known.- The data is approximately normally distributed.
What are the Steps to Perform a Z-Test?
1. State the Hypotheses: Define the null hypothesis (H0) and the alternative hypothesis (H1). - H0: There is no significant difference (e.g., the sample mean is equal to the population mean). - H1: There is a significant difference (e.g., the sample mean is not equal to the population mean).
2. Choose the Significance Level: Common significance levels are 0.05, 0.01, and 0.10.
3. Calculate the Test Statistic: Use the formula for the Z-score: - Z = (X? - ?) / (? / ?n) - Where X? is the sample mean, ? is the population mean, ? is the population standard deviation, and n is the sample size.
4. Determine the Critical Value: Using Z-tables, find the critical value corresponding to the chosen significance level.
5. Make a Decision: Compare the calculated Z-score with the critical value. - If |Z| > critical value, reject the null hypothesis (H0). - If |Z| ? critical value, fail to reject the null hypothesis (H0).
Example of a Z-Test:
Suppose a company claims that their battery lasts an average of 500 hours (? = 500 hours). A sample of 36 batteries (n = 36) is tested, and the average battery life is found to be 495 hours (X? = 495 hours) with a known population standard deviation of 20 hours (? = 20 hours). Test the company’s claim at the 0.05 significance level.
1. State the Hypotheses: - H0: ? = 500 hours - H1: ? ? 500 hours
2. Significance Level: ? = 0.05
3. Calculate the Test Statistic: - Z = (X? - ?) / (? / ?n) - Z = (495 - 500) / (20 / ?36) - Z = -5 / (20 / 6) - Z = -5 / 3.33 - Z ? -1.50
4. Determine the Critical Value: - For ? = 0.05 (two-tailed test), the critical values are approximately ±1.96.
5. Make a Decision: - Since -1.50 does not exceed ±1.96, we fail to reject the null hypothesis.
Conclusion:There is not enough evidence to reject the company's claim that the average battery life is 500 hours at the 0.05 significance level.
Use the P-value method to perform the hypothesis test for the indicated exercise. Exercise 22
Use the P-value method to perform the hypothesis test for the indicated exercise. Exercise 21
Use the P-value method to perform the hypothesis test for the indicated exercise. Exercise 20
Use the P-value method to perform the hypothesis test for the indicated exercise. Exercise 19
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