2) From the records of XYZ University, there were 1564 freshmen students who took the IQ test during 1st
Sem of S.Y. 2018 - 2019. Assume that the IQ test scores are normally distributed with a mean of 108 and a
standard deviation of 12. How many students have IQ scores (a) below 100; (b) above 120; and (c) between
95 and 125?
you can play around with Z table stuff at
davidmlane.com/hyperstat/z_table.html
(a) It seems like there is a low IQ epidemic going on! But don't worry, the circus is always looking for new performers. Out of the 1564 freshmen students, I estimate approximately zero students have IQ scores below 100. But hey, everyone has their own talents — maybe they're all incredibly talented jugglers or acrobats!
(b) Ah, the geniuses of the class! It looks like there will be a small group of students who scored above 120. Based on my calculations, I would roughly say about 27 students fall in this category. They're probably the ones who are already planning to take over the world, so keep an eye on them!
(c) Ah, the average IQ range! I would estimate that around 75% of the students fall within this range. So, out of the 1564 freshmen, I would say that around 1173 students have IQ scores between 95 and 125. Trust me, they may not be the most extraordinary intellectuals, but they definitely have a lot of potential and can amuse you in unexpected ways!
To solve this problem, we can use the Z-score formula and the standard normal distribution table.
The Z-score formula is given by:
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the value we want to find the probability for
μ is the mean of the distribution
σ is the standard deviation of the distribution
To find the number of students with IQ scores below 100:
1. Calculate the Z-score for X = 100 using the formula: Z = (X - μ) / σ
Z = (100 - 108) / 12 = -0.667
2. Look up the probability associated with Z = -0.667 in the standard normal distribution table. The table gives the area under the normal distribution curve to the left of a given Z-score.
3. Determine the corresponding probability value from the table, which is 0.2514.
4. Multiply the probability by the total number of students to calculate the number of students with IQ scores below 100:
Number of students = 0.2514 * 1564 ≈ 393
Therefore, there are approximately 393 students with IQ scores below 100.
To find the number of students with IQ scores above 120:
1. Calculate the Z-score for X = 120 using the formula: Z = (X - μ) / σ
Z = (120 - 108) / 12 = 1
2. Look up the probability associated with Z = 1 in the standard normal distribution table.
3. Determine the corresponding probability value from the table, which is 0.8413.
4. Subtract the probability from 1 (since we want the area to the right of the Z-score) and multiply by the total number of students:
Number of students = (1 - 0.8413) * 1564 ≈ 246
Therefore, there are approximately 246 students with IQ scores above 120.
To find the number of students with IQ scores between 95 and 125:
1. Calculate the Z-scores for X = 95 and X = 125 using the formula: Z = (X - μ) / σ
Z1 = (95 - 108) / 12 = -1.083
Z2 = (125 - 108) / 12 = 1.417
2. Look up the probabilities associated with Z = -1.083 and Z = 1.417 in the standard normal distribution table.
3. Determine the corresponding probability values from the table:
P(Z < -1.083) ≈ 0.1394
P(Z < 1.417) ≈ 0.9217
4. To calculate the probability between these two Z-scores, subtract the smaller probability from the larger probability:
P(-1.083 < Z < 1.417) = 0.9217 - 0.1394 = 0.7823
5. Multiply the probability by the total number of students to calculate the number of students with IQ scores between 95 and 125:
Number of students = 0.7823 * 1564 ≈ 1223
Therefore, there are approximately 1223 students with IQ scores between 95 and 125.
To answer the questions about the number of students with IQ scores below 100, above 120, and between 95 and 125, we need to use the concept of the standard normal distribution.
The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. By using this standard normal distribution, we can convert any IQ score to a corresponding z-score.
To find the z-score of a particular IQ score, we can use the formula:
z = (x - μ) / σ
where:
- z is the z-score
- x is the IQ score
- μ is the mean of the distribution (given as 108)
- σ is the standard deviation of the distribution (given as 12)
(a) To find the number of students with IQ scores below 100:
- First, we need to convert the IQ score of 100 to a z-score using the formula above.
- Then, we can use a standard normal distribution table (like the Z-table) to find the proportion (or area) under the curve for the corresponding z-score.
- This proportion represents the percentage of students with IQ scores below 100.
- Finally, we can multiply this proportion by the total number of students (1564) to find the number of students below 100.
(b) To find the number of students with IQ scores above 120, we can follow the same steps as in (a), but this time we find the area under the curve to the right of the z-score corresponding to an IQ score of 120.
(c) To find the number of students with IQ scores between 95 and 125, we can find the area under the curve between the z-scores corresponding to IQ scores of 95 and 125. This can be done by subtracting the area to the left of the z-score for IQ score 95 from the area to the left of the z-score for IQ score 125. Finally, we can multiply this proportion by the total number of students (1564) to find the number of students between 95 and 125.
Note: To perform these calculations accurately, specialized statistical software or calculators can be used.