Answer:
Explanation:
The question is incomplete. The complete question is:
Television viewing reached a new high when the global information and measurement company reported a mean daily viewing time of 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 6 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 5% of all television viewing households (to 2 decimals)?
(c.) What is the probability that a household views television more than 5 hours a day (to 4 decimals)?
Solution:
Let x be the random variable representing the television viewing times per household. Since it is normally distributed and the population mean and population standard deviation are known, we would apply the formula,
z = (x - µ)/σ
Where
x = sample mean
µ = population mean
σ = standard deviation
From the information given,
µ = 8.35
σ = 2.5
a) the probability that a household views television between 6 and 8 hours a day is expressed as
P(6 ≤ x ≤ 8)
For x = 6,
z = (6 - 8.35)/2.5 = - 0.94
Looking at the normal distribution table, the probability corresponding to the z score is 0.1736
For x = 8
z = (8 - 8.35)/2.5 = - 0.14
Looking at the normal distribution table, the probability corresponding to the z score is 0.4443
Therefore,
P(6 ≤ x ≤ 8) = 0.4443 - 0.1736 = 0.2707
b) the top 5% means greater than 95%. It means that the sample mean is greater than the population mean and the z score is positive. The corresponding z score from the normal distribution table is 1.645. Therefore,
(x - 8.35)/2.5 = 1.645
Cross multiplying, it becomes
x - 8.35 = 2.5 × 1.645 = 4.11
x = 4.11 + 8.35 = 12.46
c) the probability that a household views television more than 5 hours a day is expressed as
P(x > 5) = 1 - P(x ≤ 5)
For x = 5
z = (5 - 8.35)/2.5 = - 1.34
Looking at the normal distribution table, the probability corresponding to the z score is 0.0901
Therefore,
P(x > 5) = 1 - 0.0901 = 0.9099