The graph depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the probability that a member selected at random is from the shaded area of the graph.
To find the probability, we need to determine the area under the normal distribution curve that corresponds to the shaded region of the graph. This can be done by calculating the Z-scores for the lower and upper boundaries of the shaded area, and then using the Z-table (also known as the standard normal distribution table) to find the probability.
1. Determine the Z-scores:
The Z-score represents the number of standard deviations a value is above or below the mean in a normal distribution. We can calculate the Z-scores for the lower and upper boundaries of the shaded area using the formula:
Z = (X - μ) / σ
where:
X = the value being considered
μ = the mean of the distribution (100 in this case)
σ = the standard deviation of the distribution (15 in this case)
Let's say the lower boundary of the shaded area on the graph has an IQ score of 85, and the upper boundary has an IQ score of 115. We can calculate the Z-scores as follows:
Z_lower = (85 - 100) / 15 = -1
Z_upper = (115 - 100) / 15 = 1
2. Use the Z-table:
The Z-table provides the cumulative probability for each Z-score. It tells us what proportion of the data falls below a given Z-score. Since we are interested in the shaded area, which represents the probability of selecting a member from the sampled population within the specified range, we need to find the cumulative probability for the lower and upper Z-scores.
Looking up the Z-score of -1 in the Z-table, we find that the cumulative probability is 0.1587.
Similarly, looking up the Z-score of 1 in the Z-table, we find that the cumulative probability is 0.8413.
3. Calculate the probability:
To find the probability of selecting a member at random from the shaded area, we need to subtract the cumulative probability of the lower boundary from the cumulative probability of the upper boundary:
P = cumulative probability (Z_upper) - cumulative probability (Z_lower)
P = 0.8413 - 0.1587
P = 0.6826
Therefore, the probability that a member selected at random is from the shaded area of the graph is 0.6826, or 68.26%.