Question

A recent survey found that 54.7% of the population own their homes. In a random sample of 150 heads of households, 92 responded that they owned their homes. At 0.01 level of significance does that indicate a difference from the national proportion. 

Answer 1

Sample proportion: $\hat p=\frac{92}{150}=0.613.$

Null hypothesis $H_0:\pi=0.547.$

Alternative hypothesis $H_a:\pi\ne0.547.$

Test statistic: $z=\frac{\hat p-\pi}{\sqrt{\frac{\pi(1-\pi)}{n}}}=\frac{0.613-0.547}{\sqrt{\frac{0.547(1-0.547)}{150}}}=1.63.$

P-value: $p=2P(Z>1.63)=0.1027.$

Since the p-value is greater than 0.01, fail to reject the null hypothesis.

There is no sufficient evidence that there is a difference from the national proportion.