A random sample of n = 25 IQ scores is selected from a population (IQ scores: mean = 100, standard deviation = 15). What is the probability that the sample mean will be less than 103?
The Correct Answer and Explanation is:
To calculate the probability that the sample mean will be less than 103 for a random sample of 25 IQ scores, we can use the sampling distribution of the sample mean. The steps are as follows:
Step 1: Identify the known values
- Population mean (μ) = 100
- Population standard deviation (σ) = 15
- Sample size (n) = 25
- Sample mean (M) = 103
Step 2: Calculate the standard error of the sample mean
The standard error (SE) of the sample mean is calculated by the formula:SE=σnSE = \frac{\sigma}{\sqrt{n}}SE=nσ
Substitute the known values:SE=1525=155=3SE = \frac{15}{\sqrt{25}} = \frac{15}{5} = 3SE=2515=515=3
Step 3: Standardize the sample mean
To find the probability that the sample mean is less than 103, we need to standardize the sample mean (i.e., convert it into a z-score). The formula for the z-score is:z=M−μSEz = \frac{M – \mu}{SE}z=SEM−μ
Substitute the values:z=103−1003=33=1z = \frac{103 – 100}{3} = \frac{3}{3} = 1z=3103−100=33=1
Step 4: Find the probability corresponding to the z-score
The z-score of 1 corresponds to a cumulative probability from the standard normal distribution. Using a z-table or a normal distribution calculator, we find:P(z<1)=0.8413P(z < 1) = 0.8413P(z<1)=0.8413
Conclusion:
Thus, the probability that the sample mean is less than 103 is 0.8413 or 84.13%.
Explanation:
This means that, given a random sample of 25 IQ scores, there is an 84.13% chance that the sample mean will be less than 103. This is based on the properties of the normal distribution and how the sample mean behaves relative to the population mean, with larger sample sizes resulting in a more accurate estimate of the population mean.
