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$X$ is a normally distributed random variable with mean $49$ and standard deviation $25$ . $\newline$ What is the probability that $X$ is between $24$ and $74$ $?$ $\newline$ Use the $0.68-0.95-0.997$ rule and write your answer as a decimal. Round to the nearest thousandth if necessary. $\newline$ ____

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Find probabilities using the normal distribution I

Full solution

Q.

X

is a normally distributed random variable with mean

49

and standard deviation

25

.

\newline

What is the probability that

X

is between

24

and

74

?

\newline

Use the

0.68-0.95-0.997

rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

\newline

____

Step $1$ : Calculate z-scores for $X = 24$ and $X = 74$ : We have a normally distributed random variable $X$ with a mean ( $\mu$ ) of $49$ and a standard deviation ( $\sigma$ ) of $25$ . We want to find the probability that $X$ is between $24$ and $74$ . The $X = 74$ $0$ - $X = 74$ $1$ - $X = 74$ $2$ rule, also known as the empirical rule, states that for a normal distribution, approximately $X = 74$ $3$ of the data falls within one standard deviation of the mean, $X = 74$ $4$ within two standard deviations, and $X = 74$ $5$ within three standard deviations.
Step $2$ : Interpret the z-scores: First, we calculate the z-scores for $X = 24$ and $X = 74$ . The z-score is given by the formula $z = \frac{X - \mu}{\sigma}$ . For $X = 24$ , the z-score is $z = \frac{24 - 49}{25} = -1$ . For $X = 74$ , the z-score is $z = \frac{74 - 49}{25} = 1$ .
Step $3$ : Apply the empirical rule: Since the z-scores for $X = 24$ and $X = 74$ are $-1$ and $1$ , respectively, this means that the values $24$ and $74$ are one standard deviation below and above the mean. According to the empirical rule, approximately $68\%$ of the data in a normal distribution falls within one standard deviation of the mean.
Step $4$ : Determine the probability: Therefore, the probability that $X$ is between $24$ and $74$ is approximately $68\%$ , or $0.68$ as a decimal.
Step $5$ : Round the probability if necessary: We round the probability to the nearest thousandth if necessary. However, in this case, the probability is already at the desired precision, so no rounding is needed.

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