Question

The highway fuel economy of a 2016 Lexus RX 350 FWD 6-cylinder 3.5-L automatic 5-speed using premium fuel is a normally distributed random variable with a mean of μ = 20.00 mpg and a standard deviation of σ = 2.00 mpg.

(a) What is the standard error of X¯¯ , the mean from a random sample of 36 fill-ups by one driver? (Round your answer to 4 decimal places.)
  
Standard error of   X¯¯ = ___

(b) Within what interval would you expect the sample mean to fall, with 99 percent probability? (Round your answers to 4 decimal places.)
  
The interval is from ___ to ___

Answer 1

Solution:-

Given that,

mean = $\mu$ = 20.00

standard deviation = $\sigma$ = 2.00

a) n = 36

$\mu$ $\bar x$ =  $\mu$ = 20.00

$\sigma$ $\bar x$ = $\sigma$ / $\sqrt$ n = 2.00 / $\sqrt$ 36 = 0.3333

b) Using standard normal table,

P( -z < Z < z) = 99%

= P(Z < z) - P(Z <-z ) = 0.99

= 2P(Z < z) - 1 = 0.99

= 2P(Z < z) = 1 + 0.99

= P(Z < z) = 1.99 / 2

= P(Z < z) = 0.995

= P(Z < 2.58 ) = 0.995

= z  ± 2.58

Using z-score formula  

$\bar x$ = z * $\sigma$ $\bar x$ + $\mu$ $\bar x$   

$\bar x$ = -2.58 *0.3333 + 20.00

$\bar x$ = 19.1401

Using z-score formula  

$\bar x$ = z * $\sigma$ $\bar x$ + $\mu$ $\bar x$   

$\bar x$ = 2.58 *0.3333 + 20.00

$\bar x$ = 20.8599

The interval is from 19.1401 to 20.8599