Using the 68-95-99.7rule. Assume that a set of test scores is normally distributed with a mean of 80 and a standard deviation of 25. Use the 68-95-99.7 rule to find the following quantities. percentage of scores greater than 105 is
a score of 105 is one standard deviation above the mean
the rule says that 68% of a normally distributed population lies within one standard deviation of the mean...half above the mean and half below
so 34% are within one standard deviation above the mean
this means that 16% are above that value
Assume that a set of test scores is normally distributed with a mean of 110 and a standard deviation of 20. Use the 68-95-99.7 rule to find the following quantities.
a. The percentage of scores less than is
nothing110%=
(Round to one decimal place as needed.)
Well, if the scores are normally distributed with a mean of 80 and a standard deviation of 25, we can use the 68-95-99.7 rule to estimate the percentage of scores that are greater than 105.
According to the 68-95-99.7 rule, approximately 68% of the scores fall within one standard deviation of the mean, approximately 95% fall within two standard deviations, and approximately 99.7% fall within three standard deviations.
To find the percentage of scores greater than 105, we need to calculate how many standard deviations away 105 is from the mean.
First, we find the z-score using the formula: z = (x - μ) / σ, where x is the value we're interested in, μ is the mean, and σ is the standard deviation.
z = (105 - 80) / 25
z = 1
So, 105 is one standard deviation away from the mean.
Now, we can determine the percentage of scores greater than 105.
Approximately 68% of the scores fall within one standard deviation, which means that 100% - 68% = 32% of the scores fall outside of one standard deviation.
Since 105 is one standard deviation away from the mean, the percentage of scores greater than 105 is approximately 32%.
To find the percentage of scores greater than 105 using the 68-95-99.7 rule, we need to determine the area under the normal distribution curve to the right of 105.
1. Calculate the z-score:
The z-score is a measure of how many standard deviations an observation is above or below the mean. It is calculated using the formula:
z = (X - μ) / σ
where X is the observation, μ is the mean, and σ is the standard deviation.
In this case, X = 105, μ = 80, and σ = 25.
z = (105 - 80) / 25 = 1
2. Determine the area to the right of the z-score:
The 68-95-99.7 rule states that within one standard deviation of the mean, approximately 68% of the data falls. In this case, since we want to find the percentage greater than 105, we need to look to the right of one standard deviation, which accounts for the remaining 32% of the data.
3. Calculate the percentage:
To find the area under the normal distribution curve, you can use statistical tables or a calculator with a normal distribution function. Using a standard normal distribution table, you can find that the percentage to the right of a z-score of 1 is approximately 15.87%.
Therefore, the percentage of scores greater than 105 is approximately 15.87%.