Question

5. For a random sample of 50 households with gas stoves monitored over a week, the sample mean CO2 level (ppm) in the households was -654.16, and the sample standard deviation was s 164.43. a. Check if the CLT (central limit theorem) applies and show the two assumptions that you check. b. Calculate a 95% (two-sided) confidence interval for true average CO2 level in the population of households with gas stoves. c. Interpret your CI from b. d. Suppose the investigators made a rough guess of s 175 before collecting data. What sample size would be necessary to obtain an interval width of 50 ppm for a confidence level of 95%?

Answer 1

a)

since, sample is large enough (n≥30) , so population from which sample are taken can be approx normally distributed

so, CLT theorem applies.

b)

sample std dev ,    s =    164.43
Sample Size ,   n =    50
Sample Mean,    x̅ =   654.16

Level of Significance ,    α =    0.05          
degree of freedom=   DF=n-1=   49          
't value='   tα/2=   2.0096   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   164.430   / √   50   =   23.2539
margin of error , E=t*SE =   2.0096   *   23.254   =   46.730
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    654.16   -   46.730   =   607.430
Interval Upper Limit = x̅ + E =    654.16   -   46.730   =   700.890
confidence interval is (   607.43   < µ <   700.89   )  

c)

we can be 95% confident that true average CO2 level in the population of households with gas stoves lies within confidence interval

d)

Standard Deviation ,   σ =    175                  
sampling error ,    E = 50/2 = 25                  
Confidence Level ,   CL=   95%                  
                          
alpha =   1-CL =   5%                  
Z value =    Zα/2 =    1.960   [excel formula =normsinv(α/2)]              
                          
Sample Size,n = (Z * σ / E )² = (   1.960   *   175   /   25   ) ² =   188.2
                          
                          
So,Sample Size needed=       189