The Wechsler Adult Intelligence Scale (WAIS) is a common "IQ test" for adults. The distribution of WAIS scores for persons over 16 yrs of age is approximately Normal with mean (100) and Standard Deviation 15,
(a) Whatis the probability that a random chosen individual has a WAIS score of 105 or higher?
(b) What are the mean and standard deviation of the average WAIS score of x for a SRS of 60 people?
(C) What is the Probability that the average WAIS score of an SRS of 60 people is 105 or higher?
Use z-scores and the normal distribution table for these problems.
For example:
a) Use z-score formula: z = (x - mean)/sd
z = (105 - 100)/15 = 0.33
Look at a normal distribution table using z = 0.33 and determine the area under the curve that would represent this z-score and higher (upper tail of the distribution curve). That would then be your probability.
If a sample size is given, that needs to be computed into the formula:
z = (x - mean)/(sd/sqrt of sample size)
Note: sqrt = square root.
I hope this will get you started.
理解理解
(a) To find the probability that a randomly chosen individual has a WAIS score of 105 or higher, we need to find the area under the normal distribution curve to the right of 105. To do this, we can use z-scores.
First, calculate the z-score using the formula: z = (x - mean) / sd, where x is the given score, mean is the mean score, and sd is the standard deviation.
In this case, x = 105, mean = 100, and sd = 15.
z = (105 - 100) / 15 = 0.33
Next, we look up the z-score of 0.33 in the standard normal distribution table. The table provides the cumulative probability to the left of the z-score, so to find the probability to the right of the z-score (i.e., the probability of a score of 105 or higher), we subtract the cumulative probability from 1.
Looking up the z-score of 0.33 in the table, we find that the cumulative probability to the left of this score is approximately 0.6293.
So, the probability of a randomly chosen individual having a WAIS score of 105 or higher is 1 - 0.6293 = 0.3707, or 37.07%.
(b) To find the mean and standard deviation of the average WAIS score of x for a simple random sample (SRS) of 60 people, we use the formula:
mean of x-bar = mean
standard deviation of x-bar = sd / sqrt(n)
Here, n = 60 (sample size), mean = 100, and sd = 15.
mean of x-bar = 100
standard deviation of x-bar = 15 / sqrt(60) ≈ 1.94
So, the mean of the average WAIS score for a sample of 60 people is 100, and the standard deviation is approximately 1.94.
(c) To find the probability that the average WAIS score of an SRS of 60 people is 105 or higher, we can again use z-scores.
Using the same formula as before with n = 60:
z = (x - mean) / (sd / sqrt(n))
In this case, x = 105, mean = 100, sd = 15, and n = 60.
z = (105 - 100) / (15 / sqrt(60)) = 2.74
Next, we look up the z-score of 2.74 in the standard normal distribution table. The table provides the cumulative probability to the left of the z-score, so to find the probability to the right of the z-score (i.e., the probability of an average score of 105 or higher), we subtract the cumulative probability from 1.
Looking up the z-score of 2.74 in the table, we find that the cumulative probability to the left of this score is approximately 0.9973.
So, the probability that the average WAIS score of an SRS of 60 people is 105 or higher is 1 - 0.9973 = 0.0027, or 0.27%.
To solve each part of the problem, we'll need to calculate z-scores and use the normal distribution table.
(a) To find the probability that a randomly chosen individual has a WAIS score of 105 or higher, we'll need to calculate the z-score using the formula:
z = (x - mean) / sd
where x is the value of interest, mean is the mean of the distribution (100), and sd is the standard deviation (15).
z = (105 - 100) / 15 = 0.33
We can then look up this z-score in the normal distribution table to find the area under the curve that represents this z-score and higher. This area is the probability we're looking for.
(b) To find the mean and standard deviation of the average WAIS score for a simple random sample (SRS) of 60 people, we'll use the formula:
z = (x - mean) / (sd / sqrt(sample size))
Here, x is the mean of the distribution (100), sd is the standard deviation (15), and the sample size is 60.
The mean of the sample means (average WAIS score) will be the same as the mean of the population, which is 100.
The standard deviation of the sample means (average WAIS score) is calculated by dividing the standard deviation of the population by the square root of the sample size:
standard deviation of sample means = sd / sqrt(sample size)
= 15 / sqrt(60)
(c) To find the probability that the average WAIS score of an SRS of 60 people is 105 or higher, we'll use the same formula as in part (b):
z = (x - mean) / (sd / sqrt(sample size))
Here, x is the value of interest (105), mean is the mean of the population (100), sd is the standard deviation of the population (15), and the sample size is 60.
We'll calculate the z-score using these values and then look up the area under the curve that represents this z-score and higher in the normal distribution table. This area is the probability we're looking for.
Remember to convert the z-scores into probabilities using the normal distribution table. I hope this explanation helps you solve the problem. If you have any more questions, feel free to ask!