Z-Statistic Calculation A Step-by-Step Guide

In the realm of statistical hypothesis testing, the z-statistic serves as a crucial tool for determining the likelihood of a sample mean deviating significantly from the population mean. This article delves into the intricacies of calculating the z-statistic, employing a practical example to illustrate the process. Our focus is on Destiny's hypothesis test, where we aim to decipher the correct representation of the z-statistic given her data. We will explore the fundamental concepts underpinning hypothesis testing, the formula for calculating the z-statistic, and a detailed walkthrough of the calculations involved in Destiny's scenario. This comprehensive guide will equip you with the knowledge to confidently tackle similar statistical problems.

Understanding Hypothesis Testing

At its core, hypothesis testing is a method used to evaluate a claim or hypothesis about a population based on sample data. In hypothesis testing, we begin by formulating two competing hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis, often denoted as H0, represents the default assumption or the status quo. It's a statement that we aim to disprove. On the other hand, the alternative hypothesis, denoted as H1 or Ha, proposes a different scenario or effect that we are trying to find evidence for. The goal is to determine whether there is sufficient evidence from the sample data to reject the null hypothesis in favor of the alternative hypothesis. This decision-making process involves calculating a test statistic, such as the z-statistic, and comparing it to a critical value or calculating a p-value. The p-value represents the probability of observing a sample mean as extreme as, or more extreme than, the one obtained, assuming the null hypothesis is true. A small p-value (typically less than a predetermined significance level, often 0.05) suggests strong evidence against the null hypothesis, leading to its rejection. Conversely, a large p-value indicates that the observed sample data is consistent with the null hypothesis, and we fail to reject it. The z-statistic is particularly useful when dealing with large sample sizes or when the population standard deviation is known. It measures how many standard deviations the sample mean is away from the population mean. A large absolute value of the z-statistic suggests a significant difference between the sample mean and the population mean. In the context of Destiny's hypothesis test, we need to carefully consider the given information, including the population mean, population standard deviation, sample size, and sample mean, to accurately calculate the z-statistic and draw meaningful conclusions about the population.

The Z-Statistic Formula

The z-statistic formula is a cornerstone of statistical hypothesis testing, providing a standardized measure of the difference between a sample mean and a population mean. This formula allows us to assess the likelihood of observing a particular sample mean if the null hypothesis about the population mean is true. The z-statistic is calculated by subtracting the population mean (µ) from the sample mean (x̄), and then dividing the result by the standard error of the mean. The standard error of the mean is calculated by dividing the population standard deviation (σ) by the square root of the sample size (n). Mathematically, the formula for the z-statistic is expressed as: z = (x̄ - µ) / (σ / √n). Each component of the formula plays a crucial role in determining the final value of the z-statistic. The sample mean (x̄) represents the average value of the data collected from the sample. The population mean (µ) is the average value of the entire population, which is often a known or hypothesized value. The population standard deviation (σ) measures the spread or variability of the population data. The sample size (n) represents the number of observations in the sample. By incorporating these elements into the formula, the z-statistic effectively quantifies the discrepancy between the sample and the population, taking into account the variability within the population and the size of the sample. A larger absolute value of the z-statistic indicates a greater difference between the sample mean and the population mean, suggesting stronger evidence against the null hypothesis. Conversely, a smaller absolute value of the z-statistic suggests that the sample mean is closer to the population mean, providing less evidence to reject the null hypothesis. In the context of Destiny's hypothesis test, correctly applying this formula is paramount to determining the accurate z-statistic and interpreting the results of her test.

Calculating the Z-Statistic for Destiny's Data

To accurately determine the z-statistic for Destiny's hypothesis test, we need to meticulously apply the z-statistic formula using the provided data. Destiny is conducting a hypothesis test on a population with a known population mean (µ) of 115 and a population standard deviation (σ) of 30. Her sample consists of 25 observations (n = 25), and the sample mean (x̄) is calculated to be 120.4. Now, let's systematically plug these values into the z-statistic formula: z = (x̄ - µ) / (σ / √n). Substituting the given values, we get: z = (120.4 - 115) / (30 / √25). First, we calculate the difference between the sample mean and the population mean: 120.4 - 115 = 5.4. Next, we calculate the standard error of the mean by dividing the population standard deviation by the square root of the sample size: 30 / √25 = 30 / 5 = 6. Finally, we divide the difference between the means by the standard error: z = 5.4 / 6 = 0.9. Therefore, the z-statistic for Destiny's data is 0.9. This value represents the number of standard errors the sample mean (120.4) is away from the population mean (115). A positive z-statistic indicates that the sample mean is greater than the population mean. To interpret this z-statistic in the context of Destiny's hypothesis test, we would compare it to a critical value or calculate the p-value. The critical value is determined by the chosen significance level (alpha) and the type of test (one-tailed or two-tailed). The p-value represents the probability of observing a z-statistic as extreme as, or more extreme than, 0.9, assuming the null hypothesis is true. By comparing the p-value to the significance level, Destiny can make a decision about whether to reject or fail to reject the null hypothesis.

Detailed Breakdown of the Calculation

To ensure a clear understanding of the z-statistic calculation for Destiny's hypothesis test, let's break down each step in detail. The formula we're using is: z = (x̄ - µ) / (σ / √n), where: x̄ is the sample mean, µ is the population mean, σ is the population standard deviation, and n is the sample size. Step 1: Identify the given values. From the problem statement, we have: x̄ = 120.4 (sample mean), µ = 115 (population mean), σ = 30 (population standard deviation), n = 25 (sample size). Step 2: Calculate the difference between the sample mean and the population mean. This step involves subtracting the population mean from the sample mean: 120. 4 - 115 = 5.4. This difference represents the raw deviation of the sample mean from the population mean. Step 3: Calculate the standard error of the mean. The standard error of the mean is a measure of the variability of sample means around the population mean. It is calculated by dividing the population standard deviation by the square root of the sample size: σ / √n = 30 / √25. First, we find the square root of the sample size: √25 = 5. Then, we divide the population standard deviation by this value: 30 / 5 = 6. The standard error of the mean is 6. Step 4: Calculate the z-statistic. Now we have all the components needed to calculate the z-statistic. We divide the difference between the means (calculated in Step 2) by the standard error of the mean (calculated in Step 3): z = 5.4 / 6 = 0.9. Therefore, the z-statistic for Destiny's data is 0.9. This value indicates that the sample mean is 0.9 standard errors above the population mean. Step 5: Interpret the z-statistic. A z-statistic of 0.9 suggests that the sample mean is not drastically different from the population mean, as it is less than 1 standard error away. To make a formal decision about the hypothesis, we would need to compare this z-statistic to a critical value or calculate the p-value associated with it. This detailed breakdown clarifies the step-by-step process of calculating the z-statistic, ensuring a thorough understanding of the methodology and the meaning of the result.

Determining the Correct Z-Statistic and Conclusion

Based on our comprehensive calculation, the z-statistic for Destiny's data is 0.9. This value accurately reflects the standardized difference between the sample mean and the population mean, taking into account the variability within the population and the sample size. Now, let's revisit the original question: "Which of the following correctly depicts the z-statistic for this data?" Among the options provided, the correct answer is 0.9, which is not listed in the original options. Therefore, the provided options seem to be incorrect. It's crucial to verify the accuracy of provided options in statistical problems to avoid misinterpretations. The z-statistic of 0.9 signifies that the sample mean of 120.4 is 0.9 standard errors above the population mean of 115. To draw a definitive conclusion about Destiny's hypothesis test, we would need to compare this z-statistic to a critical value or calculate the p-value associated with it. For instance, if we were conducting a two-tailed test at a significance level of 0.05, the critical values would be approximately ±1.96. Since our calculated z-statistic of 0.9 falls within this range (-1.96 < 0.9 < 1.96), we would fail to reject the null hypothesis. This implies that there is insufficient evidence to conclude that the sample mean is significantly different from the population mean. Alternatively, we could calculate the p-value associated with a z-statistic of 0.9. The p-value represents the probability of observing a z-statistic as extreme as, or more extreme than, 0.9, assuming the null hypothesis is true. For a two-tailed test, we would calculate the area in both tails of the standard normal distribution beyond ±0.9. This p-value would be greater than 0.05, further supporting our decision to fail to reject the null hypothesis. In conclusion, the correct z-statistic for Destiny's data is 0.9. This value, along with a comparison to critical values or the calculation of a p-value, allows us to make informed decisions about hypothesis testing.

In summary, this article has provided a detailed exploration of calculating the z-statistic in the context of hypothesis testing. We meticulously analyzed Destiny's hypothesis test, demonstrating the step-by-step process of applying the z-statistic formula. We began by understanding the fundamental principles of hypothesis testing, emphasizing the importance of formulating null and alternative hypotheses. We then delved into the z-statistic formula, dissecting each component and its role in determining the final value. Through a clear and concise breakdown, we calculated the z-statistic for Destiny's data, arriving at the value of 0.9. This calculation involved identifying the given values (sample mean, population mean, population standard deviation, and sample size), calculating the difference between the means, determining the standard error of the mean, and finally, applying the z-statistic formula. Furthermore, we discussed the interpretation of the z-statistic in the context of hypothesis testing, highlighting the need to compare it to critical values or calculate the p-value to make informed decisions about rejecting or failing to reject the null hypothesis. We also addressed a potential discrepancy in the provided options, underscoring the importance of verifying the accuracy of information in statistical problem-solving. By providing a comprehensive guide, this article equips readers with the necessary knowledge and skills to confidently calculate and interpret the z-statistic in various hypothesis testing scenarios. The ability to accurately calculate the z-statistic is a crucial skill for students, researchers, and professionals who rely on statistical analysis to draw meaningful conclusions from data. This knowledge empowers individuals to make data-driven decisions and contribute to advancements in their respective fields. As statistical methods continue to play an increasingly important role in various disciplines, a solid understanding of concepts like the z-statistic will remain invaluable.