Mathematics: The proportion of eligible voters in the next election who will vote for the incumbent is assumed to be 53.6%. What is the probability that in a random sample of 470 voters, less than 48.1% say they will vote for the incumbent

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The proportion of eligible voters in the next election

SupremeDiaz17

06.02.2022

The required probability = 0.0084

Step-by-step explanation:

GIven that:

Sample proportion = 53.6% = 0.536

Sample size = 470

Then; the mean $\mu$ = n × p

mean $\mu$ = 470 × 0.536

mean $\mu$ = 251.92

Standard deviation = $\sqrt{n\times p(1-p)}$

Standard deviation = $\sqrt{470 \times 0.536(1-0.536)}$

Standard deviation = $\sqrt{470 \times 0.536(0.464)}$

Standard deviation = $\sqrt{116.89088}$

Standard deviation = 10.81

The sample mean $\overline x = n \times \dfrac{48.1}{100}$

The sample mean $\overline x = 470 \times \dfrac{48.1}{100}$

The sample mean $\overline x$ = 226.07

Thus;

$P( \overline x < 226.07) = P(Z < \dfrac{\overline x - \mu }{\sigma})$

$P( \overline x < 226.07) = P(Z < \dfrac{226.07 - 251.92 }{10.81})$

$P( \overline x < 226.07) = P(Z < -2.39)$

$P( \overline x < 226.07)$ = 0.0084

Therefore, the required probability = 0.0084

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