An intelligence test used in a particular country has scores which are normally distributed with mean 100 and standard deviation 15.In a randomly selected group of 500 people sitting the test, estimate how many have a score
i.higher than 140
ii.between 85 and 90
In my days, we had to calculate stuff like "z-scores" and look at tables in the back of text books.
Now-a-days, there are webpages like
http://davidmlane.com/normal.html
that makes these type of questions almost trivial.
An intelligence test used in a particular country has scores which are normally distributed with mean 100 and standard deviation 15.In a randomly selected group of 500 people sitting the test, estimate how many have a score
i.higher than 140
ii.between 85 and 90
To estimate the number of people with a score higher than 140, we need to calculate the proportion of scores above 140 in the normal distribution.
First, we need to standardize the value of 140 using the formula:
Z = (X - μ) / σ
where Z is the standardized score, X is the score we want to standardize (140 in this case), μ is the mean (100), and σ is the standard deviation (15).
Z = (140 - 100) / 15
Z = 40 / 15
Z ≈ 2.67
Next, we need to find the proportion of scores above a Z-score of 2.67 using a standard normal distribution table or calculator. The table or calculator will give us the area under the curve to the left of the Z-score.
Looking up a Z-score of 2.67 in the table, we find that the area to the left is approximately 0.9961.
To estimate the number of people with a score higher than 140 in the randomly selected group of 500, we multiply the proportion by the sample size:
Estimated number = Proportion * Sample size
Estimated number = 0.9961 * 500
Estimated number ≈ 498.05
Therefore, an estimated 498 people would have a score higher than 140 in the randomly selected group of 500.
To estimate the number of people with scores between 85 and 90, we need to calculate the proportion of scores falling within that range.
Firstly, we need to standardize the values of 85 and 90 using the same formula as before:
For 85:
Z = (85 - 100) / 15
Z = -15 / 15
Z = -1
For 90:
Z = (90 - 100) / 15
Z = -10 / 15
Z = -0.67
Next, we need to find the area under the curve between these two Z-scores. To do this, we calculate the difference between the areas to the left of each Z-score.
Looking up a Z-score of -1 in the table, we find that the area to the left is approximately 0.1587.
Looking up a Z-score of -0.67 in the table, we find that the area to the left is approximately 0.2514.
To find the area between these two Z-scores, we subtract the smaller area from the larger area:
Area between -1 and -0.67 = 0.2514 - 0.1587
Area between -1 and -0.67 ≈ 0.0927
To estimate the number of people with scores between 85 and 90 in the randomly selected group of 500, we multiply the proportion by the sample size:
Estimated number = Proportion * Sample size
Estimated number = 0.0927 * 500
Estimated number ≈ 46.35
Therefore, an estimated 46 people would have scores between 85 and 90 in the randomly selected group of 500.