Question

"What do you think is the ideal number of children for a family
to have?" A poll asked this question of 1011 randomly chosen
adults. Almost half (49%) thought two children was ideal. Suppose
that *p* = 0.49 is exactly true for the population of all
adults. The polling company announced a margin of error of ±3
percentage points for this poll. What is the probability that the
sample proportion *p̂* for an SRS of size *n* = 1011
falls between 0.46 and 0.52? You see that it is likely, but not
certain, that polls like this give results that are correct within
their margin of error. (Round your answer to four decimal
places.)

Answer #1

Answer)

P = 0.49

N = 1011

First we need to check the conditions of normality that is if n*p and n*(1-p) both are greater than 5 or not

N*p = 495.39

N*(1-p) = 515.61

Both the conditions are met so we can use standard normal z table to estimate the probability

z = (oberved p - claimed p)/standard error

Standard error = √{claimed p*(1-claimed p)/√n

P(0.46<p<0.52) = P(p<0.52) - P(p<0.46)

P(p<0.52)

Here observed P = 0.52, Claimed P = 0.49, N = 1011.

After substitution,

Z = 1.91

From z table, P(z<1.91) = 0.9719

P(p<0.46) = P(z<-1.91) = 0.0281

Required probability is 0.9719 - 0.0281 = 0.9438

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"What do you think is the ideal number of children for a family
to have?" A Gallup Poll asked this question of 1016 randomly chosen
adults. Almost half (49%) thought two children was ideal.† We are
supposing that the proportion of all adults who think that two
children is ideal is p = 0.49.
What is the probability that a sample proportion p̂ falls
between 0.46 and 0.52 (that is, within ±3 percentage points of the
true p) if the...

"What do you think is the ideal number of children for a family
to have?" A Gallup Poll asked this question of 1016 randomly chosen
adults. Almost half (49%) thought two children was ideal.† We are
supposing that the proportion of all adults who think that two
children is ideal is p = 0.49.
What is the probability that a sample proportion p̂ falls
between 0.46 and 0.52 (that is, within ±3 percentage points of the
true p) if the...

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