point(s) possible
Assume a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded region of the graph. Assume the variable $\mathrm{x}$ is normally distributed.
The probability that the member selected at random is from the shaded area of the graph is $\square$. (Round to four decimal places as needed.)
26
Solution
Step 1 ：We are given a normal distribution with a mean (\(\Delta\)) of 21.3 and a standard deviation (\(\Sigma\)) of 5.2. We are asked to find the probability that a member selected at random falls within the shaded region of the graph, which is between 26 and 32.
Step 2 ：To solve this, we first standardize the values of 26 and 32 using the formula for z-score, which is \((X - \text{mean}) / \text{standard deviation}\).
Step 3 ：Calculating the z-scores for 26 and 32, we get \(z1 = 0.9038461538461536\) and \(z2 = 2.0576923076923075\) respectively.
Step 4 ：Next, we use the standard normal distribution table (or a function that gives the cumulative distribution function of the standard normal distribution) to find the probabilities corresponding to these z-scores.
Step 5 ：The probabilities corresponding to \(z1\) and \(z2\) are \(prob1 = 0.8169615077083663\) and \(prob2 = 0.9801901603710788\) respectively.
Step 6 ：The probability that the member falls within the shaded region is the difference between these two probabilities, which is \(prob = 0.16322865266271247\).
Step 7 ：Rounding to four decimal places, the final answer is \(\boxed{0.1632}\).
From Solvely APP
Solution
Step 1 ：We are given a normal distribution with a mean (\(\Delta\)) of 21.3 and a standard deviation (\(\Sigma\)) of 5.2. We are asked to find the probability that a member selected at random falls within the shaded region of the graph, which is between 26 and 32.
Step 2 ：To solve this, we first standardize the values of 26 and 32 using the formula for z-score, which is \((X - \text{mean}) / \text{standard deviation}\).
Step 3 ：Calculating the z-scores for 26 and 32, we get \(z1 = 0.9038461538461536\) and \(z2 = 2.0576923076923075\) respectively.
Step 4 ：Next, we use the standard normal distribution table (or a function that gives the cumulative distribution function of the standard normal distribution) to find the probabilities corresponding to these z-scores.
Step 5 ：The probabilities corresponding to \(z1\) and \(z2\) are \(prob1 = 0.8169615077083663\) and \(prob2 = 0.9801901603710788\) respectively.
Step 6 ：The probability that the member falls within the shaded region is the difference between these two probabilities, which is \(prob = 0.16322865266271247\).
Step 7 ：Rounding to four decimal places, the final answer is \(\boxed{0.1632}\).
