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5.1.3 Statistical Reasoning Unit 5 Lesson 1 – Empirical Rule Practice Name: __________________________ Date: ____________ Period: _______ For each of the dot plots below, use the Empirical Rule to estimate the mean and the standard deviation of each of the following distributions. Then, use your calculator to determine the mean and standard deviation of each of the distributions. Did the empirical rule give you a good estimate of the standard deviation? 1. Estimate: Mean: Standard Deviation: Actual: Mean: Standard Deviation: Now that you know what the actual mean and standard deviation, calculate one standard deviation below the mean and one standard deviation above the mean. µ = _________________________ (Mean) Locate these numbers on the dot plot above. σ = ____________________________ (Standard Deviation) How many dots are between these numbers?_________ Is this close to 68%?________ Do you think that the empirical rule should apply to this distribution?_____ Why? 2. Estimate: Mean: Standard Deviation: Actual: Mean: Standard Deviation: Now that you know what the actual mean and standard deviation, calculate one standard deviation below the mean and one standard deviation above the mean. µ = _________________________ σ = ____________________________ (Mean) (Standard Deviation) Locate these numbers on the dot plot above. How many dots are between these numbers?_______ Is this close to 68%?_______ Do you think that the empirical rule should apply to this distribution?_______ Why? 3. Estimate: Mean: Standard Deviation: Actual: Mean: Standard Deviation: Now that you know what the actual mean and standard deviation, calculate one standard deviation below the mean and one standard deviation above the mean. µ = _________________________ σ = ____________________________ (Mean) (Standard Deviation) Locate these numbers on the dot plot above. How many dots are between these numbers?_____ Is this close to 68%?_______ Do you think that the empirical rule should apply to this distribution?_______ Why? 4. Estimate: Mean: Standard Deviation: Actual: Mean: Standard Deviation: Now that you know what the actual mean and standard deviation, calculate one standard deviation below the mean and one standard deviation above the mean. µ = _________________________ (Mean) σ = ____________________________ (Standard Deviation) Locate these numbers on the dot plot above. How many dots are between these numbers?_______ Is this close to 68%?_______ Do you think that the empirical rule should apply to this distribution?_______ Why? 5. Estimate: Mean: Standard Deviation: Actual: Mean: Standard Deviation: Now that you know what the actual mean and standard deviation, calculate one standard deviation below the mean and one standard deviation above the mean. µ = _________________________ (Mean) σ = ____________________________ (Standard Deviation) Locate these numbers on the dot plot above. How many dots are between these numbers?_______ Is this close to 68%?_______ Do you think that the empirical rule should apply to this distribution?_______ Why? 6. Estimate: Mean: Standard Deviation: Actual: Mean: Standard Deviation: Now that you know what the actual mean and standard deviation, calculate one standard deviation below the mean and one standard deviation above the mean. µ = _________________________ (Mean) σ = ____________________________ (Standard Deviation) Locate these numbers on the dot plot above. How many dots are between these numbers?_______ Is this close to 68%?_____ Do you think that the empirical rule should apply to this distribution?_______ Why? 7. For which distributions did you give a good estimate of the standard deviation based on the empirical rule? 8. Which distributions did not give a good estimate of the standard deviation based on the empirical rule? 9. Which distributions had close to 68% of the data within one standard deviation of the mean? What do they have in common? 10. For which type of distributions do you think the Empirical rule applies? (Describe the shape of the graph.) As you discovered, the empirical rule does not work unless your data is bell-shaped. However, not all bell-shaped graphs are normal. The next two dot plots are bell-shaped graphs. You will apply the empirical rule to determine if the bell-shaped graph is normal or not. 11. µ = _________________________ (Mean) σ = ____________________________ (Standard Deviation) Mark the mean on your dot plot above. Calculate one standard deviation above and below the mean. Calculate μ + σ __________________________________ and μ – σ_____________________________________ Mark these points on the x-axis of the dot plot. How many data points are between these values? Calculate two standard deviations below and above the mean. Calculate μ + 2σ ________________________________ and μ – 2σ_____________________________________ Mark these points on the x-axis of the dot plot. How many data points are between these values? Calculate three standard deviations below and above the mean. Calculate μ + 3σ ________________________________ and μ – 3σ_____________________________________ Mark these points on the x-axis of the dot plot. How many data points are between these values? Is it likely that this sample is from a normal population? Explain your thinking. Outliers are values that are beyond two standard deviations from the mean in either direction. Which values from the data would be considered to be outliers? ______________________ 12. µ = _________________________ (Mean) σ = ____________________________ (Standard Deviation) Mark the mean on your dot plot above. Calculate one standard deviation above and below the mean. Calculate μ + σ __________________________________ and μ – σ_____________________________________ Mark these points on the x-axis of the dot plot. How many data points are between these values? Calculate two standard deviations below and above the mean. Calculate μ + 2σ ________________________________ and μ – 2σ_____________________________________ Mark these points on the x-axis of the dot plot. How many data points are between these values? Calculate three standard deviations below and above the mean. Calculate μ + 3σ ________________________________ and μ – 3σ_____________________________________ Mark these points on the x-axis of the dot plot. How many data points are between these values? Is it likely that this sample is from a normal population? Explain your thinking. Outliers are values that are beyond two standard deviations from the mean in either direction. Which values from the data would be considered to be outliers? ______________________ Based on your observations when is it appropriate to conclude that a data set is approximately normal?