My research claim is that the population average expenditure

My research claim is that the population average expenditure on smart phones per year per person, , is different from (i.e. not equal to) $300. A random sample of 100 individuals had a sample mean of $298 and a sample standard deviation of $10.

a. State your null and alternative hypotheses. What is your test statistic formula and what is its value? Calculate the p-value and state your interpretation of the p-value.

b. Construct the 95% confidence interval for .

c. Does the interval contain 300? How is this result related to what you found in b.?

Solution

a)

Formulating the null and alternative hypotheses,              
              
Ho:   u   =   300  
Ha:    u   =/   300   [HYPOTHESES]

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As we can see, this is a    two   tailed test.      
              
              
Getting the test statistic, as              
              
X = sample mean =    298          
uo = hypothesized mean =    300          
n = sample size =    100          
s = standard deviation =    10          
              
Thus, z = (X - uo) * sqrt(n) / s =    -2   [TEST STATISTIC]

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Also, the p value is              
              
p =    0.045500264   [P VALUE]

Hence,   if the null hypothesis is true, the probabilty of getting this sample or farther from the hypothesized mean is 0.0455.

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b)

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    298          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    10          
n = sample size =    100          
              
Thus,              
Margin of Error E =    1.959963985          
Lower bound =    296.040036          
Upper bound =    299.959964          
              
Thus, the confidence interval is              
              
(   296.040036   ,   299.959964   ) [ANSWER]

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c)
  
NO. As 300 is not inside this interval, there at 0.05 level, there is sufficient evidnece that the mean expenditure on smart phones per year per person, , is different from $300.