1 Sample Proportion Hypothesis Z-Test Example

Calculations by Professor S. Gramlich

Updated 3/22/09

8-3 #6 (from Triola, “Elementary Statistics,” 10^th ed, C2006, p. 414)

Survey of Workers

In a survey of 703 randomly selected workers, 15.93% got their jobs through

Newspaper ad (based on data from Taylor Nelson Sofres Intereach). Consider a

Hypothesis test that uses a 0.05 significance level to test the claim that less than 20%

of workers get their jobs through newspaper ads.

Given

n = 703

p-hat = 15.93% = .1593

alpha = .05

Z-test

H0: P >= .20

H1: P < .20 (orig claim)

Left 1 tailed test

Q= 1-P = 1- .2 = .8

(b) What is the critical value?

cv: Zc = -1.645 for alpha=.05 from table A-2.1

(a) What is the test statistic?

ts: Z = (phat – P)/ sqrt(P*Q/n)

= (.1593 - .2)/ sqrt(.2*.8/703)

= -0.0407/ 0.015086285728094283983973179842987

~= -2.70

(c) What is the P-value?

P-val =.0035 for Z = -2.7 from table A-2.1

(d) What is the conclusion?

Conclusion: Since (P-val =.0035) < (alpha = .05), Reject H0.

or Since ts inside critcal region (Z= -2.7) < (Zc = -1.645), Reject H0.

There is sufficient evidence to support claim that less than 20% of workers get job

through newspaper.

(e) Based on the preceding results, can we conclude that 15.93% is significantly

less than 20% for all such hypothesis tests? Why or why not?

No, not for ALL Hypothesis tests because it depends on sample size and significance level.

A more strict significance level or smaller sample could cause fail to reject H0.