Let's go through each question step by step:

Question 3:

Given that $x$ is a normally distributed variable with a mean of $60$ and a standard deviation of $10$, find the following probabilities:

a) $P(x > 60)$

Since $x$ follows a normal distribution, the probability of $x$ being equal to the mean (60) is exactly 0.5. The area to the right of 60 in a standard normal distribution (which has a mean of 0) is also 0.5.

So: $P(x > 60) = 0.5$

b) $P(x < 72)$

First, we calculate the z-score for 72: $z = \frac{72 - 60}{10} = 1.2$

We then use the standard normal distribution table to find $P(z < 1.2)$. The value for $z = 1.2$ is approximately 0.8849.

Thus: $P(x < 72) = 0.8849$

c) $P(65 < x < 82)$

We need to find two z-scores:

1. For $x = 65$: $z_1 = \frac{65 - 60}{10} = 0.5$ From the standard normal table, $P(z < 0.5) \approx 0.6915$.

2. For $x = 82$: $z_2 = \frac{82 - 60}{10} = 2.2$ From the table, $P(z < 2.2) \approx 0.9861$.

Now, to find $P(65 < x < 82)$: $P(65 < x < 82) = P(z < 2.2) - P(z < 0.5) = 0.9861 - 0.6915 = 0.2946$

Question 4:

The waiting time $x$ for a lecture to begin follows a normal distribution with a mean of $3.7$ minutes and a standard deviation of $1.4$ minutes.

a) Probability that a randomly selected course has to wait for less than 2.0 minutes.

Calculate the z-score for 2.0: $z = \frac{2.0 - 3.7}{1.4} = -1.21$

Using the standard normal table, $P(z < -1.21) \approx 0.1131$.

So: $P(x < 2.0) = 0.1131$

b) Probability that a randomly selected course has to wait between 5.0 and 7.5 minutes.

1. For $x = 5.0$: $z_1 = \frac{5.0 - 3.7}{1.4} = 0.93$ From the table, $P(z < 0.93) \approx 0.8238$.

2. For $x = 7.5$: $z_2 = \frac{7.5 - 3.7}{1.4} = 2.71$ From the table, $P(z < 2.71) \approx 0.9966$.

Now, to find $P(5.0 < x < 7.5)$: $P(5.0 < x < 7.5) = P(z < 2.71) - P(z < 0.93) = 0.9966 - 0.8238 = 0.1728$

Summary:

• 3a) $P(x > 60) = 0.5$
• 3b) $P(x < 72) = 0.8849$
• 3c) $P(65 < x < 82) = 0.2946$
• 4a) $P(x < 2.0) = 0.1131$
• 4b) $P(5.0 < x < 7.5) = 0.1728$

Would you like further explanations or details on any of these parts?

Here are 5 related questions for further exploration:

1. How do you calculate the z-score for a given value in a normal distribution?
2. What is the significance of the standard normal distribution table in solving these problems?
3. How do you interpret probabilities in a normal distribution?
4. How would the probabilities change if the standard deviation was smaller or larger?
5. What happens to the probability $P(x > a)$ as $a$ increases beyond the mean?

Tip: Always visualize the normal distribution curve when solving probability questions; it helps to understand the area under the curve corresponding to the probabilities you seek.