In: Statistics and Probability
A company hopes to improve customer satisfaction, setting as a goal no more than 5% negative comments. A random survey of 350 customers found only 10 with complaints. Does this provide evidence that the company has reached its goal of getting less than 5% negative comments?
a) What is the sample statistic? Use correct notation. Round to three decimal places.
b) Write the null and alternative hypothesis. (Hint: The null is 5% or 0.05. Also, think carefully about the inequality in your alternative.)
c) Find the test statistic, z. Round to two decimal places. Show your work.
d) What is the p-value? Use Statkey or your TI-84 calculator to find the p-value. Round to three decimal places if necessary.
e) At a 5% significance level, will you reject or fail to reject H0?
f) Write your conclusion in context.
Solution:
Here, we have to use one sample z test for the population proportion.
a) What is the sample statistic?
We have
x = number of items of interest = 10
n = sample size = 350
Sample statistic = p̂ = x/n = 10/350 = 0.029
b) Write the null and alternative hypothesis.
The null and alternative hypotheses for this test are given as below:
Null hypothesis: H0: The Company has reached its goal of getting not less than 5% negative comments.
Alternative hypothesis: Ha: The Company has reached its goal of getting less than 5% negative comments.
H0: p ≥ 0.05 versus Ha: p < 0.05
This is a lower tailed test.
We are given
Level of significance = α = 0.05
c) Find the test statistic, z.
Test statistic formula for this test is given as below:
Z = (p̂ - p)/sqrt(pq/n)
Where, p̂ = Sample proportion, p is population proportion, q = 1 - p, and n is sample size
x = number of items of interest = 10
n = sample size = 350
p̂ = x/n = 10/350 = 0.028571429
p = 0.05
q = 1 - p = 1 - 0.05 = 0.95
Z = (p̂ - p)/sqrt(pq/n)
Z = (0.028571429 - 0.05) / sqrt(0.05*0.95/350)
Z = -1.8394
Test statistic = -1.84
d) What is the p-value?
P-value = 0.033
(by using Ti-84 calculator)
e) At a 5% significance level, will you reject or fail to reject H0?
P-value < α = 0.05
So, we reject the null hypothesis
f) Write your conclusion in context.
There is sufficient evidence to conclude that The Company has reached its goal of getting less than 5% negative comments.