Question

Historically, the proportion of people who trade in their old car to a car dealer when purchasing a new car is 48%. Over the previous 6 months, in a sample of 115 new-car buyers, 46 have traded in their old car. To determine (at the 5% level of significance) whether the proportion of new-car buyers that trade in their old car has statistically significantly decreased; what is the critical value? (please round your answer to 2 decimal places)

Answer 1

To test $H_0: p = 0.48$ against $H_1: p < 0.48$

Here

sample proportion $\hat{p} = \frac{46}{115} = 0.4$

and sample size $n = 115$

The test statistic can be written as

$z = \frac{(\hat{p} - 0.48)}{\sqrt{\frac{0.48*(1-0.48)}{n}}}$ which under H₀ follows a standard normal distribution.

We reject H₀ at 5% level of significance if $z_{obs} <- z_{0.05}$

The critical value $z_{critical} = -z_{0.05} = -1.64485$