Internet prices for glasses are often considerably lower tha

Internet prices for glasses are often considerably lower than those at the brick and mortar stores but quality is another question. To address that issue, some optometrists ordered 154 adult pairs of spectacles from various online stores. 6 Of the 154 pairs of spectacles, 44 pairs of eye wear contained at least one lens that did not pass optical analysis testing. (a) Conduct a test of hypotheses with the intent of establishing that that the probability of not passing the optical analysis testing is greater than 0.20. Use level of significance 0.025. Z=0.500. we accept H0 at alpha = 0.025 Round your answer to three decimal places. (b) Calculate the p?value and comment. The p-value is .3085 D. So, we would have1? accepted the null hypothesis. Round your answer to four decimal places.

Solution

a)

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   <=   0.2
Ha:   p   >   0.2
As we see, the hypothesized po =   0.2      
Getting the point estimate of p, p^,          
          
p^ = x / n =    0.285714286      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.032232919      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    2.659215781   [ANSWER, Z VALUE]

As zcrit = 1.96 for a right tailed 0.025 level test, and as z > 1.96, we REJECT HO. [ANSWER]

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B)  
          
As this is a    1   tailed test, then, getting the p value,  
          
p =    0.003916139   [ANSWER]

  
As P < 0.025, we REJECT the null hypothesis. [ANSWER]