The number of people who must be surveyed if you want to be 90% confident that the sample percentage is within two percentage points of the true percentage for the population of all adults is 1691.
The true proportion of the population is not given, so we assume it to be 0.5, that is, p = 0.5.
Hence, q = 1 - p = 1 - 0.5 = 0.5.
Confidence level given = 90%.
Z-Score corresponding to 90% confidence interval (Z) = 1.645.
We are asked to find the sample size (n) when we want to be 90% confident that the sample percentage is within two percentage points of the true percentage for the population of all adults, that is, standard error (E) = 2% = 0.02.
By the formula of standard error, we know:
E = Z√[{p(1 - p)}/n].
Substituting the values, we get
0.02 = 1.645√[{0.5*0.5}/n]
or, 0.0004 = 2.706025*0.25/n
or, n = 2.706025*0.25/0.0004 = 1691.265625 ≈ 1691 {Since the sample size has to be a whole number}.
Thus, the number of people who must be surveyed if you want to be 90% confident that the sample percentage is within two percentage points of the true percentage for the population of all adults is 1691.
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