Question

Suppose that the mean daily viewing time of television is 8.35 hours per household. Use a normal probability distribution with a standard deviation of 2.5 hours to answer the following questions about daily television viewing per household.

a. What is the probability that a household views television between 4 and 8 hours a day (to 4 decimals)?
b. How many hours of television viewing must a household have in order to be in the top 4% of all television viewing households (to 2 decimal)
c. What is the probability that a household views television more than 3 hours a day (to 4 decimals)?.

Answer 1

Calculating probabilities with a normal distribution involves converting values to z-scores and using the mean and standard deviation. For ranges of values, use cumulative distribution functions. To find specific percentiles, use inverse functions to get the corresponding values in terms of hours.

Step-by-step explanation:

When working with a normal distribution, we can calculate the probabilities of different ranges of values by using the mean and standard deviation of the distribution. The normal distribution in this context is defined by a mean (μ) of 8.35 hours and a standard deviation (σ) of 2.5 hours.

1. To find the probability that a household views television between 4 and 8 hours (P(4 < X < 8)), we use the cumulative distribution function (CDF) of the normal distribution or a standard normal table, after converting the given hours to z-scores. Calculating this probability typically involves a calculator or software capable of computing normal probabilities.
2. Identifying the viewing time that puts a household in the top 4% requires finding the z-score that corresponds to the 96th percentile (since 100% - 4% = 96%) and then converting it to the actual hours using the mean and standard deviation.
3. To determine the probability that a household views television for more than 3 hours, we find P(X > 3). Like the first part, this involves finding the z-score for 3 hours and using the CDF of the normal distribution.

Note that the actual calculations depend on the use of statistical software or a calculator with statistical functions.

Answer 2

The probability that a household views television between 4 and 8 hours a day is approximately 0.3784. A household must have approximately 3.875 hours of television viewing to be in the top 4% of all television viewing households. The probability that a household views television more than 3 hours a day is approximately 0.0162.

Step-by-step explanation:

To find the probability that a household views television between 4 and 8 hours a day, we can use the normal distribution. First, we need to standardize the values using the formula z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. For x = 4, the standardized value is z = (4 - 8.35) / 2.5 = -1.38, and for x = 8, the standardized value is z = (8 - 8.35) / 2.5 = -0.14.

Using a standard normal distribution table or a calculator, we can find the probabilities corresponding to these z-values. The probability of a household viewing between 4 and 8 hours a day is:

P(-1.38 < z < -0.14) = P(z > -0.14) - P(z > -1.38)

= 1 - P(z > -0.14) - (1 - P(z > -1.38))

= P(z > -0.14) - P(z > -1.38)

Using a standard normal distribution table or a calculator, the probability is approximately 0.3784.

For part b, we need to find the z-value corresponding to a probability of 0.04. Using a standard normal distribution table or a calculator, we find that the z-value is approximately -1.75. We can use the formula z = (x - μ) / σ to find the hours of television viewing corresponding to this z-value. Solving for x, we have -1.75 = (x - 8.35) / 2.5, which gives x = -1.75 * 2.5 + 8.35. Therefore, a household must have approximately 3.875 hours of television viewing to be in the top 4% of all television viewing households.

For part c, we need to find the probability that a household views television more than 3 hours a day. We can use the same formula z = (x - μ) / σ to standardize the value. For x = 3, the standardized value is z = (3 - 8.35) / 2.5 = -2.14. Using a standard normal distribution table or a calculator, we can find the probability corresponding to this z-value, which is approximately 0.0162.