Question

Geoff has been running a restaurant in the city for many years. He knows from experience that the proportion of customers that leave tips at this restaurant is 0.55. Geoff has recently opened a restaurant in a suburban area. It seems that tipping habits are different in the suburbs: Geoff believes that the proportion of customers that tip at his new restaurant (p) may be different to 0.55. He conducts a hypothesis test to find out. The null and alternative hypotheses are: Ho: p = 0.55 Ha: p 0.55 Geoff takes a random sample of 105 customers and notes whether or not they leave a tip. In this sample, the proportion of people that leave a tip is 0.57. a) Calculate the test statistic (z). Give your answer to 2 decimal places. Z = b) Calculate the P-value. Give your answer as a decimal to 4 decimal places. P-value =

Answer 1

Solution :

This is the two tailed test .

The null and alternative hypothesis is

H₀ : p = 0 55

H_a : p $\neq$ 0.55

$\hat p$ =0.57

n = 105

P₀ = 0.55

1 - P₀ = 1 - 055 = 0.45

Test statistic = z

= $\hat p$ - P₀ / [$\sqrt$P_{0 *} (1 - P₀ ) / n]

=0.57 - 0.55 / [$\sqrt$0.55*(0.45) /105 ]

= 0.41

P(z >0.41 ) = 1 - P(z <0.41 ) =0.3409*2= 0.6818

P-value = 0.6818