Question

1. The ambient temperature in which humans are comfortable varies with culture, activity, metabolic rate, psychological state, environment and season. For most people in the U.S., the comfort zone is 68 to 78 degrees Fahrenheit. During a recent winter, a random sample of n = 5067 homeowners was selected from the Southern region of the U.S, out of a normally distributed population of approximately N = 32,141,882 homeowners. The thermostat temperature setting (in degrees Fahrenheit) was recorded for each home. The sample mean temperature was 72.1 degrees Fahrenheit, with a standard deviation of 1.55 degrees Fahrenheit. Find a 95% confidence interval for the population mean thermostat setting in degrees Fahrenheit for homeowners from the Southern region of the U.S, during winter.

1. Determine the prerequisites have been met (3 pts):

• Random sample: yes
• n ≤ 0.05 of N:
• the population is normally distributed or n ≤ 30:

2. Construct the confidence interval (4 pts):

3. Interpret the confidence interval, in a statement (1 pt):

Answer 1

a)

• Random sample: yes
• n ≤ 0.05 of N: YES

0.05N = 0.05*32141882 = 1607094 > n = 5067

• the population is normally distributed

b)

sample std dev ,    s =    1.5500
Sample Size ,   n =    5067
Sample Mean,    x̅ =   72.1000

Level of Significance ,    α =    0.05          
'   '   '          
z value=   z α/2=   1.9600   [Excel formula =NORMSINV(α/2) ]      
                  
Standard Error , SE = s/√n =   1.550   / √   5067   =   0.0218
margin of error, E=Z*SE =   1.9600   *   0.022   =   0.043
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    72.10   -   0.043   =   72.0573
Interval Upper Limit = x̅ + E =    72.10   -   0.043   =   72.1427
95%   confidence interval is (   72.06   < µ <   72.14   )

c)

we are 95% confident that true population mean of thermostat setting in degrees Fahrenheit for homeowners from the Southern region of the U.S, during winter lies within confidence interval.