Math Problem Statement
53 58 63 68 73 78 83 Test Scores Use the Empirical Rule to answer the following questions. a. The mean is 68 b. The standard deviation is 10 × c. 9 X% of the test scores are between 63 and 73. d. 68 X% of the test scores are between 58 and 78. e. 95 X % of the test scores are between 53 and 83. f. 34 % of the test scores are between 68 and 73. g. % of the test scores are less than 68. h. 6 of the test scores are less than 73.
Solution
To solve these questions using the Empirical Rule (or 68-95-99.7 Rule), we need to understand the rule’s application:
- 68% of data falls within one standard deviation of the mean.
- 95% of data falls within two standard deviations of the mean.
- 99.7% of data falls within three standard deviations of the mean.
Given the data:
- Mean (µ) = 68
- Standard deviation (σ) = 10
Let’s go step by step:
a. The mean is:
- The mean is given directly as 68.
b. The standard deviation is:
- The standard deviation is given directly as 10.
c. 9% of the test scores are between 63 and 73.
-
From the empirical rule, the range between 63 and 73 (which is one standard deviation from the mean) will contain 68% of the data. However, 9% is a much smaller proportion.
If you're asking for the percentage that is within this narrow range, you'd want to clarify whether this refers to the part of the curve between specific values, or whether 9% is given in error.
d. 68% of the test scores are between 58 and 78.
- 58 and 78 are one standard deviation below and above the mean.
- According to the empirical rule, 68% of the test scores fall between one standard deviation below and above the mean (58 and 78).
e. 95% of the test scores are between 53 and 83.
- 53 and 83 are two standard deviations away from the mean (68 ± 2 * 10 = 53 and 83).
- The empirical rule tells us that 95% of the test scores fall within two standard deviations of the mean.
f. 34% of the test scores are between 68 and 73.
- 68 is the mean, and 73 is one standard deviation above the mean.
- According to the empirical rule, 34% of the data lies between the mean (68) and one standard deviation above it (73).
g. 50% of the test scores are less than 68.
- Since the mean divides the data into two equal halves, 50% of the test scores will be less than the mean (68).
h. 6 of the test scores are less than 73.
- Given that the 68% of the scores fall between 58 and 78 (i.e., one standard deviation below and above the mean), we need more details on the total number of test scores to calculate this. Could you clarify the total number of test scores, so we can find out how many are less than 73?
Would you like further details or clarification on any of these? Here's a list of related questions:
- How do you calculate percentages in a normal distribution using the Empirical Rule?
- What if the data isn't normally distributed? What can we use instead of the Empirical Rule?
- Can the Empirical Rule be applied to small datasets, or is it only for large datasets?
- How do you determine the exact percentage of data within specific ranges outside the 68-95-99.7 rule?
- What is the relationship between the z-score and the Empirical Rule?
Tip: When applying the Empirical Rule, always remember that it's an approximation for normally distributed data. For data that isn't normally distributed, you may need to use different methods.
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Math Problem Analysis
Mathematical Concepts
Statistics
Empirical Rule
Normal Distribution
Formulas
Empirical Rule: 68-95-99.7 Rule
Theorems
Empirical Rule
Suitable Grade Level
Grades 9-12
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