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variable with mean $29$ and standard deviation $25$ . What is the probability that $X$ is between $4$ and $54$ ?

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The Central Limit Theorem

Full solution

Q. variable with mean

29

and standard deviation

25

. What is the probability that

X

is between

4

and

54

?

Convert to Z-scores: First, we need to convert the $X$ values to Z-scores. $Z = \frac{(X - \text{mean})}{\text{standard deviation}}$
Calculate Z-score for $X=4$ : Calculate the Z-score for $X = 4$ . $Z = \frac{4 - 29}{25}$ $Z = -1$
Calculate Z-score for $X=54$ : Calculate the Z-score for $X = 54$ . $Z = \frac{54 - 29}{25}$ $Z = 1$
Look up in Z-table: Now, we look up the Z-scores in the standard normal distribution table. $\newline$ The probability for $Z = -1$ is approximately $0.1587$ . $\newline$ The probability for $Z = 1$ is approximately $0.8413$ .
Find probability between $4$ and $54$ : To find the probability that $X$ is between $4$ and $54$ , subtract the smaller probability from the larger one. $\newline$ Probability( $X$ is between $4$ and $54$ ) = $P(Z < 1) - P(Z < -1)$ $\newline$ Probability( $X$ is between $4$ and $54$ ) = $0.8413 - 0.1587$ $\newline$ Probability( $X$ is between $4$ and $54$ ) = $0.6826$

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Question

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\newline

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\newline

1

\newline

g(c)=-3

for at least one

c

-1

4

\newline

g(c)=3

for at least one

c

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1

\newline

g(c)=-3

for at least one

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1

\newline

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