A newsletter publisher believes that less than 65% of their readers own a Rolls Royce. Is there sufficient evidence at the 0.05 level to substantiate the publisher's claim?

Assuming this complete question :"A newsletter publisher believes that 65% of their readers own a Rolls Royce. Is there sufficient evidence at the 0.05 level to refute the publisher's claim? State the null and alternative hypotheses for the above scenario. "

Solution to the problem

We need to conduct a hypothesis in order to test the claim that the true proportion is less than 0.65 or 65%:

Null hypothesis:[tex]p\geq 0.65[/tex]

Alternative hypothesis:[tex]p < 0.65[/tex]

When we conduct a proportion test we need to use the z statistic, and the is given by:

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].

For this case [tex]p_o =0.65[/tex]

n represent the sample size and [tex]\hat p[/tex] the porportion estimated from the sample data.

The significance level on this case would be [tex]\alpha=0.05[/tex]

And for this case we have a left tailed test, so then the p value ca be calculated with:

[tex]p_v =P(z<z_{calc})[/tex]

And if the [tex]p_v >\alpha[/tex] we fail to reject the null hypothesis otherwise we reject the null hypothesis at the significance level given.