Answer: To estimate the probability that fewer than 70 households in a random sample of 500 are watching the series on a regular basis, we can use the binomial probability formula:
P(X < 70) = Σ [nCx * p^x * (1 - p)^(n - x)]
where:
P(X < 70) = Probability that X is less than 70
n = Total number of trials (in this case, the sample size = 500 households)
x = Number of successes we want to find the probability for (in this case, the number of households watching the series)
p = Probability of success in a single trial (in this case, 15.4% = 0.154)
(1 - p) = Probability of failure in a single trial (1 - 0.154 = 0.846)
Now, let's calculate the probability:
P(X < 70) = Σ [500C0 * (0.154)^0 * (0.846)^500 + 500C1 * (0.154)^1 * (0.846)^499 + 500C2 * (0.154)^2 * (0.846)^498 + ... + 500C69 * (0.154)^69 * (0.846)^431]
Since calculating the summation manually can be time-consuming, we can use statistical software or a calculator with the binomial cumulative probability function to find the result directly. For example, in Python, you can use the scipy library:
n = 500
p = 0.154
x = 69
The output will give you the estimated probability that fewer than 70 households in a random sample of 500 are watching the series on a regular basis.