Television viewership hit a record when the Nielsen Company reported that households averaged 8.35 hours of viewing per day (USA Today, November 11, 2009). Assuming a normal probability distribution with a standard deviation of 2.5 hours for daily television viewing per household, we can address the following questions: A) What is the probability that a household watches TV for 5 to 10 hours per day? B) How many hours must a household watch TV to fall into the top 3% of viewers? C) What is the probability that a household watches TV for more than 3 hours per day?

To solve all these questions, we will use properties of the normal probability distribution. A normal distribution is characterized by its mean (average) and its standard deviation (which measures the spread of the distribution). It is symmetric and has a bell-shaped curve.

A) The probability that a household watches TV for 5 to 10 hours per day:

We need to find the probability that a value falls between 5 and 10 on a normal distribution with a mean of 8.35 and a standard deviation of 2.5. This can be found by calculating the Z-scores for both 5 hours and 10 hours and then using a Z-table to find the probabilities.

The Z-score is calculated as: Z = (X - μ) / σ

Where: - X is the value for which we are calculating the Z-score - μ is the mean of the distribution - σ is the standard deviation of the distribution

For 5 hours: Z = (5 - 8.35) / 2.5 = -1.34

For 10 hours: Z = (10 - 8.35) / 2.5 = 0.66

Next, use a Z-table to find the probability corresponding to each Z-score. You then find the probability between the two points by subtracting the probability for Z = -1.34 from the probability for Z = 0.66.

Assuming you have the Z-table or a calculator, find the probabilities and perform the subtraction to get the probability of watching TV between 5 and 10 hours.

B) The number of hours a household must watch TV to fall into the top 3% of viewers:

To find the number of hours corresponding to the top 3%, we want to find the Z-score that leaves 3% in the upper tail of the normal distribution. We can use a Z-table to find this. The Z-score for the top 3% is typically around +1.88 but you can confirm this with the table or statistical software.

Once we have the Z-score, we use it to find the number of hours by transforming the Z-score back to the value X using the formula:

X = Z * σ + μ

For the top 3%, we will plug in the Z-score value, the standard deviation, and the mean: X = 1.88 * 2.5 + 8.35 Then calculate X to find the answer.

C) The probability that a household watches TV for more than 3 hours per day:

To find this probability, first calculate the Z-score for 3 hours, then find the corresponding probability from the Z-table.

For 3 hours: Z = (3 - 8.35) / 2.5 = -2.14

Next, we find the cumulative probability for Z = -2.14. Since the normal distribution is symmetric, we can find the probability for watching less than 3 hours and then subtract it from 1 to find the probability for more than 3 hours.

Using a Z-table or statistical software, find the value for Z = -2.14 and subtract that value from 1 to get the probability for more than 3 hours of watching TV.

Extra:

Understanding these concepts might be easier with a bit more background knowledge.

The normal distribution is a probability distribution that is symmetric around the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It's one of the key concepts in statistics and is often used to represent real-world variables that tend to cluster around an average value.

The Z-score is a statistical measure that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. A Z-score of 0 indicates that the data point's score is identical to the mean score, while a Z-score of 1.0 indicates a value that is one standard deviation from the mean. Z-scores may also be negative, which indicates a value less than the mean.

A Z-table is used in statistics to find probabilities for a statistical measurement that follows a standard normal distribution. The Z-table helps you find that probability or percentile without complex calculations.

The probability of an event lies between 0 (the event never happens) and 1 (the event always happens). When we calculate probabilities using a normal distribution, we are often determining the likelihood that a data point falls within a certain range of values.

Calculating such probabilities and finding cut-off points within a distribution (like the top 3% in question B) are essential tools for statisticians when analyzing data and making decisions based on statistical evidence.