Final answer:
To sketch the normal curve and show +3 and -3 standard deviations and the percentages using the empirical rule, calculate the z-scores and use a z-table to find the percentages. The probability that raw score x < 87 cannot be determined using the empirical rule. The percentage of raw data values between 2 and 10 can be found using the z-scores and a z-table. The number of raw data values more than 67 is 0 when the sample size is n = 82.
Step-by-step explanation:
To sketch the normal curve and show +3 and -3 standard deviations and the percentages using the empirical rule:
A. The normal curve is a bell-shaped curve that represents a normal distribution. The mean of the population is 6 and the standard deviation is 2. We can calculate the values for +3 and -3 standard deviations using the formula: +3 standard deviations = mean + (3 * standard deviation) = 6 + (3 * 2) = 12 and -3 standard deviations = mean - (3 * standard deviation) = 6 - (3 * 2) = 0. The percentages using the empirical rule would be approximately 99.7% between -3 standard deviations and +3 standard deviations, approximately 95% between -2 standard deviations and +2 standard deviations, and approximately 68% between -1 standard deviation and +1 standard deviation.
B. To calculate the probability that the raw score x < 87 using the empirical rule, we need to find out the number of standard deviations away from the mean 87 is. The formula to calculate z-score is z = (x - mean) / standard deviation = (87 - 6) / 2 = 40.5. Since the z-score is not within +3 standard deviations or -3 standard deviations, the probability cannot be determined using the empirical rule.
C. To find the percentage of raw data values between 2 and 10 using the empirical rule, we need to calculate the z-scores for 2 and 10. The z-score for 2 is (2 - 6) / 2 = -2 and the z-score for 10 is (10 - 6) / 2 = 2. We can then use a z-table to find the percentage between -2 and 2 which is approximately 95%.
D. If the sample size is n = 82, to find the number of raw data values that were more than 67, we need to calculate the z-score for 67 using the formula z = (x - mean) / standard deviation = (67 - 6) / 2 = 30.5. Using a z-table, we can find the area to the right of 30.5, which is approximately 0.00. So, the number of raw data values more than 67 is 0.