Scores on a visual perception test are normally distributed with a mean of 2020 and a standard deviation of 250.
a)If one subject is randomly selected and tested, find the probability of a score greater than 1800
b)if 50 subjects are randomly selected and tested fid the probability that the sample mean is between 2000 and 2100
a) Z = (score-mean)/SD
b) Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities related to the Z scores.
To solve these probability questions involving a normal distribution, we can use the standard normal distribution table or calculator. However, if you don't have access to those, we can also use the formula for converting a value to a z-score and then use the standard normal distribution table.
Let's go step-by-step to solve each problem:
a) To find the probability of a score greater than 1800, we need to convert this value into a z-score and then find the area to the right of that z-score on the standard normal distribution curve.
The formula to calculate the z-score is: z = (x - μ) / σ
where:
- z is the z-score
- x is the raw score
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
So, in this case:
z = (1800 - 2020) / 250
= -220 / 250
= -0.88
Now, we need to find the area to the right of this z-score (-0.88) in the standard normal distribution table or calculator. The table will give us the probability associated with this area.
Using the standard normal distribution table, we find that the probability associated with a z-score of -0.88 is approximately 0.8106.
Therefore, the probability of a score greater than 1800 is 0.8106 or 81.06%.
b) To find the probability that the sample mean is between 2000 and 2100, we need to calculate the z-scores for both of these values and then find the area between those z-scores on the standard normal distribution curve.
First, we calculate the z-score for the lower value, which is 2000:
z1 = (2000 - 2020) / (250 / √50)
= -20 / (35.3553)
= -0.56
Next, we calculate the z-score for the higher value, which is 2100:
z2 = (2100 - 2020) / (250 / √50)
= 80 / (35.3553)
= 2.26
Now, we need to find the area between these two z-scores (-0.56 and 2.26) in the standard normal distribution table or calculator.
Using the standard normal distribution table, we find that the area between these two z-scores is approximately 0.6151.
Therefore, the probability that the sample mean is between 2000 and 2100 is 0.6151 or 61.51%.