2. Why is it important to know the mean and standard deviation for a data set when applying the empirical rule?
3. If we are focused on 68% of the normal distribution, what percentage of the distribution is left in the upper tail only?
4. What value separates the 50% of the distribution from the other 50% of the distribution?
Consider this scenario for questions 5 - 8.
A standardized test was given to a set of high school juniors and the distribution of the data is bell shaped. The mean score is 800 and the standard deviation is 120.
5. Between which two scores did 95% of the students score?
6. To qualify for a special summer camp for accelerated students, a student must score within the top 16% of all scores on the test. What score must a student make to qualify for summer camp?
7. What score is 1/2 standard deviation above the mean?
8. A student scores 900 on the test. How many more points did the student need to qualify for summer camp?
2. What empirical rule? Look at the problems you are given afterward.
3. (1-.68)/2 = ? (This is assuming that the 68% are about the mean of a normal distribution.)
4. Look at the definitions of they three measures of central tendency.
5. Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.475) related to the Z scores. Insert in above equation and solve.
6. Use same table and equation.
7. Z = .5 Use equation.
8. Relate to answer on 6.
A student has a mean score of five tests taken. What score must she obtain on her next test to have a mean average of 88 on all six tests?
2. It is important to know the mean and standard deviation for a data set when applying the empirical rule because they are used to determine the range within which a certain percentage of the data values will fall. The mean tells us the central tendency or average value of the data set, while the standard deviation measures the dispersion or spread of the data points around the mean. Using these values, we can apply the empirical rule to make probabilistic statements about the data distribution.
3. Since the empirical rule states that approximately 68% of the data falls within one standard deviation of the mean, the remaining percentage in the upper tail only would be (100% - 68%) = 32%.
4. The value that separates the 50% of the distribution from the other 50% is the median. In a bell-shaped distribution, the median is also equal to the mean.
5. Using the empirical rule, we know that approximately 95% of the data falls within two standard deviations of the mean. So, to find the score range, we can calculate the scores:
Lower score = mean - (2 * standard deviation)
Upper score = mean + (2 * standard deviation)
Lower score = 800 - (2 * 120) = 560
Upper score = 800 + (2 * 120) = 1040
Therefore, between 560 and 1040, approximately 95% of the students scored.
6. To qualify for the top 16% of all scores, the student's score must fall within one standard deviation above the mean.
Score = mean + (1 * standard deviation)
Score = 800 + (1 * 120) = 920
Therefore, a student must score at least 920 to qualify for summer camp.
7. To find the score that is 1/2 standard deviation above the mean, we can calculate:
Score = mean + (0.5 * standard deviation)
Score = 800 + (0.5 * 120) = 860
Therefore, the score that is 1/2 standard deviation above the mean is 860.
8. The student scored 900 on the test. To find out how many more points the student needed to qualify for summer camp, we can calculate the difference:
Difference = Score to qualify - Student's score = 920 - 900 = 20
Therefore, the student needed 20 more points to qualify for summer camp.
2. Knowing the mean and standard deviation for a data set is important when applying the empirical rule because it allows us to understand how the data is distributed. The empirical rule states that for a bell-shaped distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. In order to determine the percentage of data that falls within a certain range, we need to know the mean and standard deviation.
3. The empirical rule states that approximately 68% of the data falls within one standard deviation of the mean. Since this is a symmetric distribution, 34% of the data falls in the upper tail and 34% falls in the lower tail. Therefore, if we are focused on the 68% region, there is (100% - 68%) / 2 = 16% left in the upper tail only.
4. The value that separates the 50% of the distribution from the other 50% is the mean. In a symmetric distribution, the mean represents the center of the distribution, and half of the data falls below the mean, while the other half falls above the mean.
Consider this scenario for questions 5 - 8:
A standardized test was given to a set of high school juniors and the distribution of the data is bell-shaped. The mean score is 800 and the standard deviation is 120.
5. To find the scores between which 95% of the students scored, we can use the empirical rule. Since 95% falls within two standard deviations of the mean, we can calculate the scores as follows:
Lower score = mean - 2 * standard deviation
Upper score = mean + 2 * standard deviation
Lower score = 800 - 2 * 120 = 800 - 240 = 560
Upper score = 800 + 2 * 120 = 800 + 240 = 1040
Therefore, 95% of the students scored between 560 and 1040.
6. To qualify for the special summer camp for accelerated students, a student must score within the top 16% of all scores on the test. This means that the student's score should be higher than 84% of all scores. Using the empirical rule, we can find the score that corresponds to the top 84% (100% - 16%) of scores:
Top score = mean + 1 * standard deviation
Top score = 800 + 1 * 120 = 800 + 120 = 920
Therefore, a student must score at least 920 to qualify for summer camp.
7. To find the score that is 1/2 standard deviation above the mean, we can calculate it as:
Score = mean + 1/2 * standard deviation
Score = 800 + 1/2 * 120 = 800 + 60 = 860
Therefore, the score that is 1/2 standard deviation above the mean is 860.
8. If a student scores 900 on the test, we can calculate how many more points the student needed to qualify for summer camp by subtracting the qualifying score (920) from their actual score:
Difference = actual score - qualifying score
Difference = 900 - 920 = -20
Since the difference is negative, it means the student scored 20 points below the qualifying score. Therefore, the student needed 20 more points to qualify for summer camp.