Suppose scores on an IQ test are normally distributed.If the test has mean of 100 and standard deviation of 10,what is the probability that a person who takes the test will score between 90 and 110?
the following website is perfectly suited for your problem
http://davidmlane.com/hyperstat/z_table.html
enter:
mean: 100
SD: 10
click on between, and set at 90 and 110
I get .682689
Ah, the magical land of probabilities! Let me calculate that for you.
For a normally distributed set of scores with a mean of 100 and a standard deviation of 10, we need to find the area under the curve between 90 and 110.
Now, if you look closely, you'll notice that the range from 90 to 110 includes the mean of 100. In a normal distribution, the mean serves as the point where the curve is perfectly balanced, like a seesaw with equally weighted clowns on both sides. So, half of the area under the curve is on one side and half on the other.
To determine the probability of scoring between 90 and 110, we add up the area under the curve on both sides. The area under the curve is the probability, like the likelihood of encountering a clown at a circus.
First, we find the z-scores for 90 and 110. The formula for calculating a z-score is (X - μ) / σ, where X is the score, μ is the mean, and σ is the standard deviation.
For 90:
z = (90 - 100) / 10 = -1
For 110:
z = (110 - 100) / 10 = 1
Then, we consult the z-table (or perhaps the z-circus in this case) to find the area between -1 and 1.
Finding the area associated with a z-score of -1 in the z-table, we get approximately 0.1587.
Finding the area associated with a z-score of 1 in the z-table, we get approximately 0.8413.
Now, we subtract the area associated with -1 from the area associated with 1:
0.8413 - 0.1587 = 0.6826
So, there is a 68.26% (approximately) probability that a person who takes the IQ test will score between 90 and 110.
Just remember, statistics can be tricky, just like a clown trying to juggle flaming torches on a unicycle.
To find the probability that a person who takes the test will score between 90 and 110, we need to calculate the area under the normal curve between these two scores.
Step 1: Standardize the scores
We need to standardize the scores using the formula
z = (x - μ) / σ
where z is the standard score, x is the raw score, μ is the mean, and σ is the standard deviation.
For a score of 90:
z1 = (90 - 100) / 10 = -1
For a score of 110:
z2 = (110 - 100) / 10 = 1
Step 2: Calculate the cumulative probability
Next, we need to find the cumulative probability associated with these two standard scores. We can use the standard normal distribution table or a calculator to find the probabilities associated with these z-scores.
Using the standard normal distribution table, we find:
P(z < -1) ≈ 0.1587 (probability associated with z1)
P(z < 1) ≈ 0.8413 (probability associated with z2)
Step 3: Calculate the probability between the two scores
To find the probability between two scores, we subtract the probability associated with the lower score from the probability associated with the higher score.
P(90 < x < 110) = P(-1 < z < 1) ≈ P(z < 1) - P(z < -1) ≈ 0.8413 - 0.1587 ≈ 0.6826
Therefore, the probability that a person who takes the test will score between 90 and 110 is approximately 0.6826, or 68.26%.
To calculate the probability that a person who takes the test will score between 90 and 110, we need to use the properties of the normal distribution. Here's how you can calculate it step by step:
Step 1: Standardize the scores
Since the scores follow a normal distribution, we need to convert them into standard units using the formula:
z = (x - μ) / σ
Where:
z is the z-score (standardized score)
x is the raw score (in this case, the score we want to find the probability for)
μ is the mean of the distribution
σ is the standard deviation of the distribution
So, for the lower boundary of x = 90:
z1 = (90 - 100) / 10 = -1
And for the upper boundary of x = 110:
z2 = (110 - 100) / 10 = 1
Step 2: Look up the z-scores in the standard normal distribution table
The standard normal distribution table provides the probabilities associated with different z-scores. Look up the probabilities corresponding to z1 and z2. Since the table typically provides probabilities for z-scores up to two decimal places, you may need to use the closest z-score entries.
Step 3: Calculate the probability
Once you have the respective probabilities for z1 and z2 from the standard normal distribution table, you can calculate the probability of a person scoring between 90 and 110 (inclusive).
P(90 ≤ x ≤ 110) = P(z1 ≤ z ≤ z2)
If the table provides probabilities for z-scores up to two decimal places, you may need to subtract the two probabilities obtained and round to two decimal places.
Keep in mind that if the table doesn't include the exact z-scores, you'll need to use the closest available ones and account for any rounding errors.
For example, if the table gives P(z ≤ -1) = 0.1587 and P(z ≤ 1) = 0.8413, the probability of scoring between 90 and 110 would be:
P(90 ≤ x ≤ 110) = P(z1 ≤ z ≤ z2) = P(z ≤ 1) - P(z ≤ -1) = 0.8413 - 0.1587 = 0.6826
So, the probability that a person who takes the IQ test will score between 90 and 110 is approximately 0.6826 or 68.26%.