Question

A survey of 100 senior citizens was undertaken and it was found that 20 of them had the flu vaccine this year. Do these data suggest that the proportion of senior citizens who get the flu vaccine is significantly less than 23%? Test at α = 0.01 α = 0.01 Round your answers to three decimal places.

(a). Without rounding any interim calculations, compute the test statistic:

(b). Compute the critical value:

(c). Using your answer from part (a), compute the p-value

Answer 1

Solution:

Given:

Sample size = n = 100

x = Number of senior citizens had the flu vaccine = 20

Level of significance = 0.01

We have to test the proportion of senior citizens who get the flu vaccine is significantly less than 23%.

Thus hypothesis of the study are:

H0: p = 0.23 Vs H1: p < 0.23

Part a) Without rounding any interim calculations, compute the test statistic:

Formula:

where

Thus

Part (b). Compute the critical value:

Level of significance = 0.01

Since this is left tailed test, look in z table for Area = 0.0100 and find corresponding z value.

Area 0.0099 is closest to 0.0100 and it corresponds to -2.3 and 0.03

thus z critical value = -2.33

Part c) Using your answer from part (a), compute the p-value

p-value = P( Z < z test statistic value)

p-value = P( Z < -0.71 )

Look in z table for z = -0.7 and 0.01 and find corresponding area.

Thus we get:

P( Z < -0.71) = 0.2389

Thus

p-value = 0.2389

Since p-value = 0.2389 > 0.01 significance level, we failed to reject H0 and thus there is not sufficient evidence to conclude that the proportion of senior citizens who get the flu vaccine is significantly less than 23%.