Final answer:
The College-level Mathematics question involves calculating the proportion of households with three cell phones and constructing confidence intervals. P' represents the proportion of households in question, and the normal distribution is applicable.
Step-by-step explanation:
The subject of the question is Mathematics, and it appears to be aimed at the College level, involving topics such as probability, random variables, confidence intervals, hypothesis testing, and distributions. A random variable, in this context, often represents some aspect of a population that we're interested in studying. Based on the information provided, the random variable P' is defined as the proportion of households that have three cell phones. The distribution that should be used for this problem is the normal distribution, as indicated by the notation ~N, which implies a normally distributed variable.
To construct a 90 percent confidence interval, one would calculate the margin of error using the standard deviation and the critical value from the z-distribution corresponding to the 90% confidence level. Increasing the confidence level to 95 percent would result in a wider confidence interval, as the critical value would be larger, reflecting the need for more certainty about the range within which the true population parameter lies.