Speaking to a group of analysts in January 2006 a brokerage

Speaking to a group of analysts in January 2006, a brokerage firm executive claimed that at least 70% of investors are currently confident of meeting their investment objectives. A UBS Investor Optimism Survey, conducted over the period January 2 to January 15, found that 67% of investors were confident of meeting their investment objectives (CNBC, January 20, 2006).

Formulate the hypotheses that can be used to test the validity of the brokerage firm executive\'s claim.

H0: p
Ha: p

Assume the UBS Investor Optimism Survey collected information from 300 investors. What is the p-value for the hypothesis test (to 4 decimals)?

At  = .05, should the executive\'s claim be rejected?

Solution

a)

As the claim is to be tested, it is the null hypothesis:

Ho: p >= 0.70
Ha: p < 0.70 [ANSWER]

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b)

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   >=   0.7
Ha:   p   <   0.7
As we see, the hypothesized po =   0.7      
Getting the point estimate of p, p^,          
          
p^ = x / n =    0.67      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.026457513      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    -1.133893419      
          
As this is a    1   tailed test, then, getting the p value,  
          
p =    0.128419629   [ANSWER, P VALUE]

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c)

As
  
significance level =    0.05      

Comparing p and the significance value, we   FAIL TO REJECT THE NULL HYPOTHESIS.      

WE DO NOT REJECT THE EXECUTIVE\'S CLAIM, THAT IS. [ANSWER]