Question

According to a research​ institution, men spent an average of ​$135.62 on​ Valentine's Day gifts in 2009. Assume the standard deviation for this population is ​$40 and that it is normally distributed. A random sample of 10 men who celebrate​ Valentine's Day was selected.

Complete parts a through e.

a. Calculate the standard error of the mean. sigma Subscript x overbarσ=

​(Round to two decimal places as​ needed.)

b. What is the probability that the sample mean will be less than ​$130?

P(x<$130)=

​(Round to four decimal places as​ needed.)

c. What is the probability that the sample mean will be more than ​$140?

P(x>$140)=

​(Round to four decimal places as​ needed.)

d. What is the probability that the sample mean will be between ​$110 and ​$165​?

P($110≤x≤$165)=

​(Round to four decimal places as​ needed.)

e. Identify the symmetrical interval that includes​ 95% of the sample means if the true population mean is ​$135.62.

≤x ≤​

​(Round to the nearest dollar as​ needed.)

Answer 1

Solution :

Given that,

mean = $\mu$ = $135.62

standard deviation = $\sigma$ = ​$40

a ) n = 10

$\mu$$\bar x$ = 135.62

$\sigma$$\bar x$  =  ($\sigma$ /$\sqrt$n) = ( ​40 / $\sqrt$10 ) = 12.65

b ) P (  $\bar x$ < 130 )

P ( $\bar x$ - $\mu$$\bar x$/$\sigma$_{$\bar x$}) < (130 - 135.62 / 12.65)

P ( z < - 5.62 /12.65 )

P ( z < - 0.44 )

Using z table

= 0.3300

Probability = 0.3300

c ) P (  $\bar x$ > 140 )

= 1 - P (  $\bar x$ < 140 )

= 1 - P ( $\bar x$ - $\mu$$\bar x$/$\sigma$_{$\bar x$}) < (140 - 135.62 / 12.65)

= 1 - P ( z < 4.38 /12.65 )

= 1 - P ( z < 0.35)

Using z table

= 1 - 0.6368

= 0.3632

Probability = 0.3632

d ) ( 110 ≤ $\bar x$ ≤ 165)

P ( 110 - 135.62 / 12.65) ≤ ( $\bar x$ - $\mu$$\bar x$/$\sigma$_{$\bar x$})   ≤ ( 165 - 135.62 / 12.65)

P ( - 25.62 / 12.65 ≤ z ≤ 29.38 / 12.65 )

P (- 2.03 ≤ z ≤ 2.32 )

P (z ≤ 2.32 ) - p ( z ≤ - 2.03 )

Using z table

= 0.9898 - 0.0212

= 0.9686

Probability = 0.9686

e ) The critical z-value = 0.05$\alpha$  = 0.05 is Z_C= 1.96 Z_C= 1.96.

symmetrical interval = ( $\bar x$ - ( Z_C * $\sigma$/ $\sqrt$n ) ,  $\bar x$ + ( Z_C * $\sigma$/ $\sqrt$n )

= ( 135.62 - 1.96 * 40 $\sqrt$10, 135.62 -+1.96 * 40 $\sqrt$10)

= (135.62 - 24.792, 135.62 + 24.792

=(110.828,160.412)

= ( 111 , 160 )

$111 ≤ x ≤ $160