Statistics II Midterm Exam2 (1)

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Statistics II Midterm Exam – 100 Marks 6:00 – 9:00 PM Worth: 25% One Sample Hypothesis Test (10 marks) Imagine you are conducting a study to determine whether the average score of students in a particular class is significantly different from the national average score of 75. You collect a random sample of 30 students from the class and find their average score to be 72 with a standard deviation of 8. Perform a one-sample hypothesis test to determine whether the class's average score differs significantly from the national average. Clearly state your null and alternative hypotheses, calculate the test statistic and p-value, and provide a conclusion based on the significance level of α = 0.05. Null hypothesis 1 H0 = 75 2 Alternative hypothesis Ha: 75 3 the critical values are – 2.064 and 2.064. 4 observed test statistics = 1.96 Conclusion there is not sufficient evidence to conclude that the average score of a group of students on standardized test is significantly different from the national average of 75 Two-Sample t Tests and Degrees of Freedom (8 marks) You are investigating whether there is a significant difference in the mean heights of two different populations: Population A and Population B. You collect a random sample of 20 individuals from Population A and find their mean height to be 65 inches with a standard deviation of 3 inches. From Population B, you collect a sample of 25 individuals and find their mean height to be 68 inches with a standard deviation of 4 inches. Conduct a two- sample t-test to determine if there is a significant difference in mean heights between the two populations. Also, calculate the degrees of freedom for this test. The two-sample t-test indicates a significant difference in mean heights between Population A and Population B

Analysis of Variance (ANOVA) and Assumptions (8 marks) Suppose you are analyzing the effectiveness of three different teaching methods (Method A, Method B, and Method C) on improving students' test scores. You collect test scores from three separate groups of students: one group for each teaching method. Conduct an analysis of variance (ANOVA) to determine whether there is a significant difference in mean test scores among the three teaching methods. Also, discuss the assumptions of ANOVA and whether they are met in this scenario. To analyze the effectiveness of three teaching methods on test score use anova. 1 Hypothesis H 0: no difference in mean test scores among methods. H 1: difference exists 2 Assumptions Independence test score is independent. Normality score within each group is normally distributed. Homogeneous of variance: variance of score in each group are equal. 3 Anova: calculate F statistics and p-value. 4 interpreting : Reject H0 if p-value less than 0.05 indicating a significant difference. 5Assumtion check : independence ensure random sampling. Normality: use histograms or normality test Homogeneity of variance: check withing levyne’s test If assumption is violated, consider non-parametric tests like Kruskal-Wallis Type 1 and Type 2 Errors (6 marks) Explain the concepts of Type 1 and Type 2 errors in hypothesis testing. Provide examples to illustrate each type of error. Type 1 error incorrectly rejecting a true null hypothesis example convicting an innocent person in a court trail. Type 2. Error” Failing to reject a false null hypothesis. Example falsely concluding a patient is healthy when they have a disease. Null Hypothesis and Alternative Hypothesis (5 marks)

Define the null hypothesis and the alternative hypothesis. Discuss their roles in hypothesis testing and provide an example of each. null hypothesis (H0): Assumes no difference or effect example the average score is 75. Alternative hypothesis (HA) suggests a difference or effect example the average score. They are used in hypothesis testing to determine if there is enough evidence to reject the null hypothesis in favor of the alternative. Central Tendency and Variability (5 marks) Define central tendency and variability in statistics. Explain how measures such as mean, median, mode, and range are used to describe central tendency and variability in a dataset. central tendency is the typical value in a dataset, while variability measures the spread of the data points. mean Average of all values, influenced by extreme values. Median Middle value when values are sorted, less affected by extremes. Model Most frequent value(s) in a dataset. Range difference between the largest and smallest value indicating data spread. A-Value and Its Representation (5 marks) What does the term "a-value" represent in hypothesis testing? Discuss its significance in decision-making during hypothesis testing and how it relates to the chosen significance level (α). The α-value is the significance level in hypothesis testing, representing the threshold for rejecting the null hypothesis. It is compared to the p-value, with a lower α-value requiring stronger evidence to reject the null hypothesis. The α-value determines the risk of Type 1 errors and guides decision-making in hypothesis testing. One-Tailed vs. Two-Tailed Test of Hypothesis (4 marks) Distinguish between a one-tailed and a two-tailed test of hypothesis. When would you use each type of test, and how does the choice of tails affect the interpretation of the results?

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In a one tailed test the critical region is in one tail, used when a specific direction is hypothesized. A two tailed test has critical regions in both tails used when no specific direction is hypothesized. One tailed test is more sensitive to detecting effects in the specified direction while two tailed tests are more conservative. Conduct a Test of Hypothesis about a Population Mean (8 Marks) Imagine you are a researcher investigating whether the average commute time to work in a city has changed from the previous year's average of 30 minutes. You collect a random sample of 50 commuters and find their average commute time to be 32 minutes with a standard deviation of 5 minutes. Conduct a test of hypothesis to determine whether there is a significant difference in the average commute time compared to the previous year. Clearly state your null and alternative hypotheses, perform the hypothesis test, and provide a conclusion based on the results. Null Hypothesis (H0): The average commute time is 30 minutes. Alternative Hypothesis (Ha): The average commute time is not 30 minutes. Using one-sample t-test with a significance level of 0.05 and 49 degrees of freedom, we find a t- value of 2.82, which exceeds the critical t-value of 2.009. Therefore, we reject the null hypothesis. Conclusion: There is enough evidence to suggest that the average commute time to work has changed significantly from the previous year's average of 30 minutes Variance (4 marks) A clothing retailer is comparing the variance in sales between two different seasons. Provide a brief explanation of how variance is calculated and why it is essential in this scenario. Use real sales data (you can invent values) for both seasons to illustrate your explanation. Variance measures the spread of data points from the mean. It's calculated by finding the average of squared differences from the mean. For the clothing retailer comparing sales between seasons, variance helps understand sales variability. For instance, if Spring sales are 100, 120, 110, 105, 115, and summer sales are 90, 130, 95, 125, 105, the variance for Spring is lower (approximately 50) than for Summer (approximately 353.2), indicating less variability in Spring sales.

Range and Tests of Hypothesis (4 marks) Explain the concept of range in statistical terms. Then, discuss how the range can be used in a hypothesis test scenario, providing an example from a real-world context. Range in statistic is the difference between the largest and smallest values in a dataset, indicating the spread of the data. In hypothesis testing is the range can be used to compare the variability between groups. For example, in a study comparing the effectiveness of two teaching methods, a larger range of scores in one group may suggest more variability in outcomes compared to the other group. Analysis of Variance and ANOVA Assumptions (4 marks) A researcher is examining the impact of three different teaching methods on student exam scores. Perform an analysis of variance (ANOVA) to determine if there is a significant difference in exam scores between the three teaching methods. Discuss the assumptions of ANOVA and check if they are met in this scenario. To determine if there's a significant difference in exam scores between the three teaching methods, the researcher can conduct an ANOVA test. The assumptions of ANOVA include independence of observations, normality of residuals, and homogeneity of variances. These assumptions can be checked using residual analysis and formal tests. If assumptions are met, ANOVA results can be interpreted to determine if there's a significant difference in exam scores. Type 1 and Type 2 Errors (6 marks) Explain the concepts of Type 1 and Type 2 errors in the context of a criminal trial. How can these errors be minimized, and what are the consequences of each type of error? Type 1 error: Convicting an innocent person (false positive). Type 2 Error: Acquitting a guilty person (false negative). To minimize these errors, ensure thorough evidence collection, provide competent legal representation, select impartial juries, and use reliable forensic techniques.

Consequences: Type 1 error leads to wrongful punishment, while Type 2 error allows a guilty person to go free, potentially endangering society and undermining justice Null Hypothesis and Alternative Hypothesis (4 marks) A pharmaceutical company is testing a new drug to reduce cholesterol levels. Formulate the null and alternative hypotheses for this study and explain the potential consequences of accepting or rejecting the null hypothesis. Null Hypothesis (H0): The new drug does not reduce cholesterol. Alternative Hypothesis (Ha): The new drug reduces cholesterol. Accepting the null hypothesis when it's false (Type 2 error) would result in missing potential health benefits. Rejecting the null hypothesis when it's true (Type 1 error) would lead to adopting an ineffective drug wasting resources. 5-Step Procedure to Test a Hypothesis (6 marks) Outline and explain the five-step procedure to test a hypothesis using an example from a business scenario. Ensure you cover all essential aspects of the process. 1 State hypothesis: Define H0 and Ha 2 Set Significance level: Choose Choose α (e.g., 0.05). 3 Collect Data: Analyze data with appropriate test. 4 Make decision: Compare p-value to α. 5 Draw Conclusion: Reject H0 if p-value ≤ α; otherwise, accept H0. Example: Scenario: Testing new product packaging’s impact on sale. Hypothesis: H0: New packaging does not increase sales. Ha: New packaging increases sales. Significance Level : α = 0.05 Data Collection : sales data for old and new packaging Test: Use t-test or ANOVA to compare sales. Decision: Reject H0 if p-value less than or equal 0.05

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Conclusion: If rejected, new packaging increases sales: if not no significant increase observed. What does a p-value represent? (7 marks) Explain the concept of a p-value and its significance in hypothesis testing. Discuss how different p-values might influence your decision to accept or reject the null hypothesis. A P-value represents represents the probability of obtaining results as extreme as the observed results, assuming that the null hypothesis is true. In hypothesis testing, a smaller p-value indicates stronger evidence against the null hypothesis, leading to its rejection in favor of the alternative hypothesis. Different p-values influence the decision to accept or reject the null hypothesis as follows.  If the p-value is less than or equal to the significance level (α), typically 0.05, the null hypothesis is rejected.  If the p-value is greater than the significance level, the null hypothesis is not rejected. Therefore, a smaller p-value provides stronger evidence against the null hypothesis, increasing the likelihood of rejecting in the favor of the alternative hypothesis.

Statistics II Midterm Exam2 (1)

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