Due: Thursday, 8 August 2024, 11:55 PMWe are given a normally distributed random variable X with µ = 75 and σ = 5.Find P (X > 85).Find k such as P(X<k) = 0.997. Interpret your result.Find the two values where the middle 68.27% of the distribution of X lie.

To find P(X > 85), we first need to convert the raw score of 85 to a z-score. The z-score is calculated as follows:

Z = (X - µ) / σ Z = (85 - 75) / 5 = 2

The z-score of 2 tells us that 85 is 2 standard deviations above the mean. We then look up this z-score in the z-table to find the proportion of scores below 85. The table tells us that 0.9772 (or 97.72%) of the scores lie below 85. Therefore, the proportion of scores above 85 is 1 - 0.9772 = 0.0228 or 2.28%. So, P(X > 85) = 0.0228.

To find k such as P(X < k) = 0.997, we first look up 0.997 in the z-table to find the corresponding z-score. The z-score is approximately 2.75. We then convert this z-score back to a raw score using the formula:

X = Zσ + µ X = 2.75 * 5 + 75 = 88.75

So, k = 88.75. This means that approximately 99.7% of the scores lie below 88.75.

The middle 68.27% of the distribution of X lies within one standard deviation of the mean. This is a property of normal distributions. Therefore, the two values are µ - σ = 75 - 5 = 70 and µ + σ = 75 + 5 = 80. So, the middle 68.27% of the distribution of X lies between 70 and 80.