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
1)higher than 140,
2)below 120,
3)between 100 and 110,
4)between 85 and 90.
To estimate the number of people with certain scores in a randomly selected group of 500 people sitting the test, we can use the standard normal distribution.
1) To estimate the number of people with scores higher than 140:
First, we need to standardize the score of 140 using the formula z = (x - μ) / σ, where x is the score, μ is the mean, and σ is the standard deviation.
z = (140 - 100) / 15
z = 2.67
Using a standard normal distribution table or calculator, we can find the proportion of individuals with a score higher than 140 (or a z-score greater than 2.67). The proportion is approximately 0.0035.
To estimate the number of people, we multiply this proportion by the sample size:
Number of people = Proportion * Sample size
Number of people = 0.0035 * 500
Number of people ≈ 1.75
Therefore, we can estimate that approximately 2 people will have a score higher than 140 in the group of 500.
2) To estimate the number of people with scores below 120:
Similarly, we standardize the score of 120:
z = (120 - 100) / 15
z = 1.33
Using the standard normal distribution table or calculator, we can find the proportion of individuals with a score below 120 (or a z-score less than 1.33). The proportion is approximately 0.908.
Number of people = Proportion * Sample size
Number of people = 0.908 * 500
Number of people ≈ 454
Therefore, we can estimate that approximately 454 people will have a score below 120 in the group of 500.
3) To estimate the number of people with scores between 100 and 110:
Standardize the lower and upper bounds of the range:
Lower z = (100 - 100) / 15
Lower z = 0
Upper z = (110 - 100) / 15
Upper z = 0.67
Using the standard normal distribution table or calculator, find the proportion of individuals between the z-scores of 0 and 0.67. The proportion is approximately 0.252.
Number of people = Proportion * Sample size
Number of people = 0.252 * 500
Number of people ≈ 126
Therefore, we can estimate that approximately 126 people will have a score between 100 and 110 in the group of 500.
4) To estimate the number of people with scores between 85 and 90:
Standardize the lower and upper bounds:
Lower z = (85 - 100) / 15
Lower z = -1
Upper z = (90 - 100) / 15
Upper z = -0.67
Using the standard normal distribution table or calculator, find the proportion of individuals between the z-scores of -1 and -0.67. The proportion is approximately 0.248.
Number of people = Proportion * Sample size
Number of people = 0.248 * 500
Number of people ≈ 124
Therefore, we can estimate that approximately 124 people will have a score between 85 and 90 in the group of 500.
To answer these questions, we need to use the standard normal distribution table or a calculator with a normal distribution function. This table or calculator can be used to find the area under the normal curve for a given z-score.
1) To estimate the number of people with a score higher than 140, we need to find the area to the right of the z-score corresponding to a raw score of 140.
First, we calculate the z-score using the formula: z = (X - μ) / σ, where X is the score, μ is the mean, and σ is the standard deviation.
So, z = (140 - 100) / 15 = 2.67 (rounded to two decimal places).
Then, using the standard normal distribution table or a calculator, we find the area to the right of 2.67. The table or calculator will give us the value of this area, which represents the proportion of the population with scores higher than 140.
2) To estimate the number of people with a score below 120, we need to find the area to the left of the z-score corresponding to a raw score of 120.
Using the same formula as above, we calculate the z-score: z = (120 - 100) / 15 = 1.33 (rounded to two decimal places).
Next, we determine the area to the left of 1.33 using the standard normal distribution table or a calculator. This area represents the proportion of the population with scores below 120.
3) To estimate the number of people with a score between 100 and 110, we need to find the area between the z-scores corresponding to these raw scores.
First, we calculate the z-scores:
- For 100: z = (100 - 100) / 15 = 0.
- For 110: z = (110 - 100) / 15 = 0.67 (rounded to two decimal places).
Next, we find the area between these two z-scores using the standard normal distribution table or a calculator. This area represents the proportion of the population with scores between 100 and 110.
4) To estimate the number of people with a score between 85 and 90, we follow the same steps as in question 3.
Calculate the z-scores:
- For 85: z = (85 - 100) / 15 = -1.
- For 90: z = (90 - 100) / 15 = -0.67 (rounded to two decimal places).
Then, find the area between these two z-scores using the standard normal distribution table or a calculator. This area represents the proportion of the population with scores between 85 and 90.
Remember, once we have the proportion of the population, we can estimate the number of people in the group by multiplying the proportion by the total number of people in the group. In this case, there are 500 people taking the test.