To find the probability that the sample mean scores will be between 85 and 125 points for a sample size of 20 adults, we can use the Central Limit Theorem. The probability is approximately 0.99996.
To find the probability that the sample mean scores will be between 85 and 125 points for a sample size of 20 adults, we can use the Central Limit Theorem. According to the Central Limit Theorem, the sampling distribution of the sample means will be approximately normal regardless of the shape of the population distribution, as long as the sample size is large enough. Since the population of scores on the Wechsler Adult Intelligence Scale is approximately normally distributed, we can use the normal distribution to solve this problem.
First, we need to calculate the standard error of the sample mean. The standard error is equal to the standard deviation of the population divided by the square root of the sample size. In this case, the standard deviation of the population is 20 and the sample size is 20, so the standard error is 20 / sqrt(20) = 4.47.
Next, we can standardize the values 85 and 125 using the formula z = (x - mean) / standard error, where x is the value, mean is the population mean, and standard error is the standard error of the sample mean. For 85, z = (85 - 105) / 4.47 = -4.47. For 125, z = (125 - 105) / 4.47 = 4.47.
Using a standard normal distribution table or a calculator, we can find the corresponding probabilities for z = -4.47 and z = 4.47. The probability that the sample mean scores will be between 85 and 125 points is the difference between these two probabilities. Let's calculate:
- Probability for z = -4.47: approximately 0.00002
- Probability for z = 4.47: approximately 0.99998
The probability that the sample mean scores will be between 85 and 125 points is approximately 0.99998 - 0.00002 = 0.99996.