To solve these probability questions for a normally distributed random variable $X$ with mean $\mu = 50$ and standard deviation $\sigma = 5.2$, we'll need to use the standard normal distribution (also known as the $Z$-distribution).

The standard normal variable $Z$ is defined as:

$Z = \frac{X - \mu}{\sigma}$

This transformation converts $X$ into a standard normal variable with mean 0 and standard deviation 1. We'll use the standard normal table (or a calculator) to find the probabilities corresponding to the calculated $Z$-scores.

Part (a): Probability $X < 55.2$

1. First, calculate the $Z$-score: $Z = \frac{55.2 - 50}{5.2} = \frac{5.2}{5.2} = 1$

2. Using the standard normal distribution table, the probability corresponding to $Z = 1$ is approximately 0.8413.

   Thus, $P(X < 55.2) \approx 0.8413$.

Part (b): Probability $X > 60.3$

1. Calculate the $Z$-score: $Z = \frac{60.3 - 50}{5.2} = \frac{10.3}{5.2} \approx 1.98$

2. The probability corresponding to $Z = 1.98$ is approximately 0.9761. Since we want $P(X > 60.3)$, we take the complement: $P(X > 60.3) = 1 - 0.9761 = 0.0239$

   Thus, $P(X > 60.3) \approx 0.0239$.

Part (c): Probability $52 < X < 57.2$

1. Calculate the $Z$-scores for $X = 52$ and $X = 57.2$:

   • For $X = 52$: $Z = \frac{52 - 50}{5.2} = \frac{2}{5.2} \approx 0.385$ The corresponding probability is approximately 0.6517.

   • For $X = 57.2$: $Z = \frac{57.2 - 50}{5.2} = \frac{7.2}{5.2} \approx 1.385$ The corresponding probability is approximately 0.9177.

2. The probability that $X$ lies between 52 and 57.2 is the difference between these two probabilities: $P(52 < X < 57.2) = 0.9177 - 0.6517 = 0.2660$

   Thus, $P(52 < X < 57.2) \approx 0.2660$.

Part (d): Probability $40 < X < 65$

1. Calculate the $Z$-scores for $X = 40$ and $X = 65$:

   • For $X = 40$: $Z = \frac{40 - 50}{5.2} = \frac{-10}{5.2} \approx -1.923$ The corresponding probability is approximately 0.0274.

   • For $X = 65$: $Z = \frac{65 - 50}{5.2} = \frac{15}{5.2} \approx 2.885$ The corresponding probability is approximately 0.9981.

2. The probability that $X$ lies between 40 and 65 is: $P(40 < X < 65) = 0.9981 - 0.0274 = 0.9707$

   Thus, $P(40 < X < 65) \approx 0.9707$.

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Would you like more details or have any questions? Here are some related questions:

1. What is the probability that $X$ takes on a value less than 45?
2. How do we interpret the standard normal distribution table?
3. What is the $Z$-score if $X$ is exactly equal to the mean $\mu = 50$?
4. What does it mean if a $Z$-score is negative?
5. How do we calculate probabilities for normal distributions without using a table?

Tip: The area under the standard normal curve between any two $Z$-scores gives the probability that the random variable lies between those two values.