9.3 Comparing two independent population proportions -- rrc math  (Page 2/20)

This is a test of two population proportions. Let M and F be the subscripts for males and females. Then p _M and p _F are the desired population proportions.

Random variable:

H ₀ : p _F = p _M   H ₀ : p _F – p _M = 0

H _a : p _F < p _M   H _a : p _F – p _M <0

The words "less than" tell you the test is left-tailed.

Distribution for the test: Since this is a test of two population proportions, the distribution is normal:

$p_c=\frac{x_F+x_M}{n_F+n_M}=\frac{156+183}{2169+2231}=\text{0}\text{.077}$
$1-p_c=0.923$
Therefore,
${p^{\prime }}_F–{p^{\prime }}_M\sim N\left(0,\sqrt{(0.077)(0.923)\left(\frac{1}{2169}+\frac{1}{2231}\right)}\right)$
p′ _F – p′ _M follows an approximate normal distribution.

Calculate the p -value using the normal distribution:
p -value = 0.1045
Estimated proportion for females: 0.0719
Estimated proportion for males: 0.082

Graph:

Decision: Since α < p -value, Do not reject H ₀

Conclusion: At the 1% level of significance, from the sample data, there is not sufficient evidence to conclude that the proportion of girls sending “sexts” is less than the proportion of boys sending “sexts.”

This is a test of two population proportions. Let W and A be the subscripts for the whites and African Americans. Then p _W and p _A are the desired population proportions.

Random variable:

H ₀ : p _W = p _A   H ₀ : p _W – p _A = 0

H _a : p _W > p _A   H _a : p _W – p _A >0

The words "more popular" indicate that the test is right-tailed.

Distribution for the test: The distribution is approximately normal:

$p_c=\frac{x_W+x_A}{n_W+n_A}=\frac{134+12}{1343+232}=0.0927$

$1-p_c=0.9073$

Therefore,

${p^{\prime }}_W–{p^{\prime }}_A\backsim N\left(0,\sqrt{\left(0.0927\right)\left(0.9073\right)\left(\frac{1}{1343}+\frac{1}{232}\right)}\right)$

${p^{\prime }}_W–{p^{\prime }}_A$ follows an approximate normal distribution.

Calculate the p -value using the normal distribution:

Graph:

Decision: Since α > p -value, reject the H ₀ .

Conclusion: At the 5% level of significance, from the sample data, there is sufficient evidence to conclude that a larger proportion of white cell phone owners use iPhones than African Americans.