Question

What is the probability, rounded to at least 4 decimal places, that in a random sample of 158 households, more than 54 but fewer than 81 of them have personal computers, based on the information that 60% of households have personal computers and assuming you round z-value calculations to 2 decimal places? Please express this probability as P(54 < X < 81)."

Answer 1

To find the probability of a specific range in a random sample of households with personal computers, you need to calculate the cumulative probability using the normal distribution and z-scores.

Step-by-step explanation:

To find the probability that in a random sample of 158 households, more than 54 but fewer than 81 of them have personal computers, we need to calculate the cumulative probability between 54 and 81. We can use the normal distribution and the z-values to calculate this probability.

1. First, calculate the z-score for 54 and 81 using the formula z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. In this case, the mean (μ) is 0.60*158 and the standard deviation (σ) is √((0.60*158)*(1-0.60)).
2. Next, use the z-table or calculator to find the cumulative probability for z < z-score(81) and subtract the cumulative probability for z < z-score(54).
3. Finally, round the probability to at least 4 decimal places.