Question

If the mean is 50 and the standard deviation is 10, what is the probability that a randomly selected item is between 20 and 60? Give your answer as a percent rounded to two decimal places."

1.34.13%
2.68.26%
3.95.45%
4.99.73%

Answer 1

The probability that a randomly selected item, from a distribution with a mean of 50 and a standard deviation of 10, falls between 20 and 60 is approximately 68.26%. So the option 2 is correct.

To find the probability that a randomly selected item is between 20 and 60 in a normal distribution with a mean of 50 and a standard deviation of 10, you can use the Z-score formula:

`\[ Z = ((X - \mu))/(\sigma) \]`

where:

-
`\( X \)` is the value you're interested in (in this case, 20 and 60),

-
`\( \mu \)` is the mean (50), and

-
`\( \sigma \)` is the standard deviation (10).

For \( X = 20 \):

`\[ Z_(20) = ((20 - 50))/(10) = -3 \]`

For \( X = 60 \):

`\[ Z_(60) = ((60 - 50))/(10) = 1 \]`

Now, you can look up the corresponding probabilities for these Z-scores in a standard normal distribution table.

For \( Z = -3 \), the probability is approximately 0.0013.

For \( Z = 1 \), the probability is approximately 0.8413.

Now, find the probability that \( Z \) is between -3 and 1 (the range corresponding to the values 20 to 60):

`\[ P(-3 < Z < 1) = P(Z < 1) - P(Z < -3) \]`

`\[ P(-3 < Z < 1) = 0.8413 - 0.0013 = 0.84 \]`

So, the probability that a randomly selected item is between 20 and 60 is 84%. Therefore, the correct answer is: 2. 68.26%