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A random variable is normally distributed with a mean of μ = 50 and a standard deviation of σ = 5. What is the probability that the random variable will assume a value between 45 and 55 (to 3 decimals)?

User Jiju John
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Final answer:

The probability that the random variable will assume a value between 45 and 55 is approximately 0.682 (or 68.2%).

Step-by-step explanation:

To find the probability that a random variable with a normally distributed mean of 50 and standard deviation of 5 will assume a value between 45 and 55, we need to calculate the area under the normal curve between these two values.

Using z-scores, we can convert the values 45 and 55 to their corresponding z-scores, which will give us the proportion of the area under the curve between these values.

By using the standard normal distribution table or a calculator, we can find that the z-scores for 45 and 55 are -1 and 1 respectively.

Therefore, the probability that the random variable will assume a value between 45 and 55 is the difference between the cumulative probabilities corresponding to these z-scores, which is approximately 0.682 (or 68.2%).

Since the standard normal distribution table provides the area to the left of a Z-score, we find the probability of Z being less than 1 and subtract the probability of Z being less than -1 to get the probability of Z being between -1 and 1.

Probability(Z < 1) - Probability(Z < -1) is approximately 0.8413 - 0.1587 = 0.6826.

So, the probability that the random variable assumes a value between 45 and 55 is 0.683 (rounded to three decimal places).

User Ian Turner
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