Question

3. According to research, 25% of adults participate in fitness activities at least twice a week. However, these fitness activities change as people get older, and occasionally participants become non-participants as they age. In a recent survey of 100 randomly selected adults, 20 indicated that they participated in a fitness 2 activity at least twice a week. At α = 0.1, can we conclude that less than 25% of adults participate in fitness activities at least twice a week?

a) State the null and alternative hypotheses in the context of the question.

b) Calculate the value of the test statistic. Check the required condition(s) to validate the test statistic.

c) Find the p-value.

d) What conclusion can we draw?

Answer 1

Solution :

This is the left tailed test .

The null and alternative hypothesis is

H₀ : p =0.25

H_a : p < 0.25

n= 100

x = 20

$\hat p$ = x / n = 20 / 100 = 0.20

P₀ = 0.25

1 - P₀ = 1 - 0.25 = 0.75

Test statistic = z

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

=0.20-0.25 / [ $\sqrt$ (0.25*0.75) / 100]

= -1.15

Test statistic = z = -1.15

P(z < -1.15 ) = 0.1251

P-value = 0.1251

$\alpha$ = 0.1

P-value > $\alpha$

0.1251 > 0.1

Fail to reject the null hypothesis .

There is not sufficient evidence to suggest that