A humane society claims that less than 32 of US households o

A humane society claims that less than 32% of U.S. households own a dog. In a random sample of 399 U.S. households,154 say they own a dog. At

alphaequals=0.03

is there enough evidence to support the society\'s claim?(a) Write the claim mathematically and identify

Upper H 0

and

Upper H Subscript aHa.

(b) Find the critical value(s) and identify the rejection region(s). (c) Find the standardized test statistic. (d) Decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

Solution

a)

Claim: p < 0.32.

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   >=   0.32
Ha:   p   <   0.32 [ANSWER]

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b) As this is a left tailed test and alpha = 0.03, the critical value is

zcrit = -1.880793608 [ANSWER, CRITICAL VALUE]

And thus, the rejection region is

z < -1.880793608. [ANSWER, REJECTION REGION]

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C)

As we see, the hypothesized po =   0.32      
Getting the point estimate of p, p^,          
          
p^ = x / n =    0.385964912      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.023353017      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    2.824684791   [ANSWER, TEST STATISTIC]

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D)

As z > -1.8808, then we FAIL TO REJECT THE NULL HYPOTHESIS.

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E)

There is no significant evidence that less than 32% of U.S. households own a dog.