A survey of 100 senior citizen
A survey of 100 senior citizens was undertaken and it was foundthat 20 of them had the flu vaccine this year. Do these datasuggest that the proportion of senior citizens who get the fluvaccine is significantly less than 23%? Test at α = 0.01 α = 0.01Round your answers to three decimal places.
(a). Without rounding any interim calculations, compute the teststatistic:
(b). Compute the critical value:
(c). Using your answer from part (a), compute the p-value
Answer:
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 theflu 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 and0.03
thus z critical value = -2.33
Part c) Using your answer from part (a), compute thep-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 correspondingarea.
Thus we get:
P( Z < -0.71) = 0.2389
Thus
p-value = 0.2389
Since p-value = 0.2389 > 0.01 significance level, we failedto reject H0 and thus there is not sufficient evidence to concludethat the proportion of senior citizens who get the flu vaccine issignificantly less than 23%.
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