X is a normally distributed random variable with a mean of 8.00. If the probability that X is less than 9.54 is 0.67, then what is the standard deviation of X?

3.5 TRY

To find the standard deviation of a normally distributed random variable, we need to use the Z-score. The Z-score measures the number of standard deviations an individual value is from the mean.

In this case, we are given that the probability that X is less than 9.54 is 0.67. Using this information, we can find the corresponding Z-score using a standard normal distribution table or calculator.

The Z-score formula is:

Z = (X - μ) / σ

Where:
- Z is the Z-score
- X is the value of the random variable
- μ is the mean of the random variable
- σ is the standard deviation of the random variable

In our case, we have:
Z = (9.54 - 8.00) / σ

We are given that the probability of X being less than 9.54 is 0.67. To find the Z-score that corresponds to this probability, we need to use the inverse cumulative distribution function (CDF) of the standard normal distribution. This function can be found in statistical software or by referring to a standard normal distribution table.

Using the inverse CDF, we find that the Z-score corresponding to a probability of 0.67 is approximately 0.43.

Now we can solve for the standard deviation, σ:
0.43 = (9.54 - 8.00) / σ

Rearranging the equation, we have:
σ = (9.54 - 8.00) / 0.43

Simplifying, we get:
σ ≈ 3.56

Therefore, the standard deviation of X is approximately 3.56.