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$X$ is a normally distributed random variable with mean $65$ and standard deviation $8$ . $\newline$ What is the probability that $X$ is between $62$ and $68$ $?$ $\newline$ Write your answer as a decimal rounded to the nearest thousandth. $\newline$ ____

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Find probabilities using the normal distribution II

Full solution

Q.

X

is a normally distributed random variable with mean

65

and standard deviation

8

.

\newline

What is the probability that

X

is between

62

and

68

?

\newline

Write your answer as a decimal rounded to the nearest thousandth.

\newline

____

Convert X values to Z-scores: To find the probability that X is between $62$ and $68$ , we first need to convert the X values to Z-scores, which are standardized scores that tell us how many standard deviations away from the mean our values are. The formula for a Z-score is $Z = \frac{(X - \mu)}{\sigma}$ , where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.
Calculate Z-score for lower bound: Let's calculate the Z-score for the lower bound, which is $X_1 = 62$ . $\newline$ Using the formula $Z = \frac{X - \mu}{\sigma}$ , we get $Z_1 = \frac{62 - 65}{8} = \frac{-3}{8} = -0.375$ .
Calculate Z-score for upper bound: Now, let's calculate the Z-score for the upper bound, which is $X_2 = 68$ . $\newline$ Using the formula $Z = \frac{X - \mu}{\sigma}$ , we get $Z_2 = \frac{68 - 65}{8} = \frac{3}{8} = 0.375$ .
Find probability for $Z_1$ and $Z_2$ : Next, we need to find the probability that $Z$ is less than $Z_1$ and $Z_2$ . We can use the standard normal distribution table or a calculator with normal distribution functions to find these probabilities. The probability for $Z_1 = -0.375$ is approximately $P(Z < -0.375) = 0.3531$ , and the probability for $Z_2 = 0.375$ is approximately $P(Z < 0.375) = 0.6463$ .
Calculate probability of X between $62$ and $68$ : To find the probability that X is between $62$ and $68$ , we need to subtract the probability of $Z_1$ from the probability of $Z_2$ . This gives us $P(62 < X < 68) = P(Z_2) - P(Z_1) = 0.6463 - 0.3531 = 0.2932$ .

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