Answer:
Step-by-step explanation:
To find the probability that the random variable x lies between 4 and 7, we can use the standard normal distribution and z-scores.
First, we need to standardize the values of 4 and 7 using the population mean and variance provided.
The formula to calculate the z-score is:
z = (x - mean) / standard deviation
In this case, the mean (μ) is 6 and the variance (σ^2) is 4. The standard deviation (σ) is the square root of the variance, which is 2 in this case.
For 4:
z1 = (4 - 6) / 2 = -1
For 7:
z2 = (7 - 6) / 2 = 0.5
Once we have the z-scores, we can look up the corresponding probabilities in the standard normal distribution table or use a calculator.
Using the standard normal distribution table, we can find the probabilities associated with the z-scores.
The probability that x lies between 4 and 7 can be calculated as the difference between the two probabilities:
P(4 < x < 7) = P(x < 7) - P(x < 4)
Using the z-scores we calculated earlier, we can find these probabilities from the standard normal distribution table.
P(x < 7) = P(z < 0.5) ≈ 0.6915 (rounded to four decimal places)
P(x < 4) = P(z < -1) ≈ 0.1587 (rounded to four decimal places)
Finally, we can calculate the probability that x lies between 4 and 7:
P(4 < x < 7) = 0.6915 - 0.1587 ≈ 0.5328 (rounded to four decimal places)
Therefore, the probability that x lies between 4 and 7 is approximately 0.5328.
