To solve this problem, we'll calculate the probability that a woman watches more than 30 hours of TV per week, using the normal distribution and the given mean and standard deviation.

Six steps to solve the problem:

1. Identify the population of interest:

• Population: Women who watch TV.
• We're asked to find the probability that a randomly selected woman watches more than 30 hours of TV per week.

2. Identify the random variable of interest:

• Let $X$ be the number of hours a woman watches TV per week.
• $X \sim N(\mu, \sigma^2)$, where $\mu = 34$ hours and $\sigma = 4.5$ hours.

3. State the probability statement:

• We're tasked with finding $P(X > 30)$, where $X$ follows the normal distribution with mean $\mu = 34$ and standard deviation $\sigma = 4.5$.

4. Convert to the standard normal distribution:

• To calculate this probability, we'll convert the normal random variable $X$ into a standard normal variable $Z$, using the formula: $Z = \frac{X - \mu}{\sigma}$ Substituting the values: $Z = \frac{30 - 34}{4.5} = \frac{-4}{4.5} \approx -0.889$

5. Use the standard normal table:

• Now, we'll find the probability $P(Z > -0.889)$.
• First, look up the probability for $Z = -0.889$ in a standard normal distribution table or use a calculator. This gives: $P(Z \leq -0.889) \approx 0.1879$
• To find $P(Z > -0.889)$, subtract this from 1: $P(Z > -0.889) = 1 - 0.1879 = 0.8121$

6. State the conclusion:

• The probability that a randomly selected woman watches more than 30 hours of TV per week is approximately 0.8121 or 81.21%.

Let me know if you'd like me to upload a step-by-step solution, including the work on paper or further clarifications.

Follow-up questions:

1. What is the probability that a woman watches less than 30 hours of TV per week?
2. How would the probability change if the average number of hours a woman watches TV increases to 36 hours?
3. What is the probability that a man watches more than 30 hours of TV per week, using the same steps?
4. If the standard deviation for women decreases to 3 hours, how does that affect the probability of watching more than 30 hours of TV?
5. How would we calculate the probability for a woman watching between 30 and 40 hours of TV per week?

Tip:

For solving probability problems involving normal distributions, always remember to convert the raw score to a standard score $Z$, which simplifies the use of probability tables or software.