Respuesta :
Answer:
(a) The sampling distribution of[tex]\overline{X}[/tex] = Population mean = 79
(b) P ( [tex]\overline{X}[/tex] greater than 81.2 ) = 0.1357
(c) P ([tex]\overline{X}[/tex] less than or equals 74.4 ) = .0107
(d) P (77.6 less than [tex]\overline{X}[/tex] less than 83.2 ) = .7401
Step-by-step explanation:
Given -
Sample size ( n ) = 81
Population mean [tex](\nu)[/tex] = 79
Standard deviation [tex](\sigma )[/tex] = 18
(a) Describe the sampling distribution of [tex]\overline{X}[/tex]
For large sample using central limit theorem
the sampling distribution of[tex]\overline{X}[/tex] = Population mean = 79
(b) What is Upper P ( [tex]\overline{X}[/tex] greater than 81.2 ) =
[tex]P(\overline{X}> 81.2)[/tex] = [tex]P(\frac{\overline{X} - \nu }{\frac{\sigma }{\sqrt{n}}}> \frac{81.2 - 79}{\frac{18}{\sqrt{81}}})[/tex]
= [tex]P(Z> 1.1)[/tex]
= [tex]1 - P(Z< 1.1)[/tex]
= 1 - .8643 =
= 0.1357
(c) What is Upper P ([tex]\overline{X}[/tex] less than or equals 74.4 ) =
[tex]P(\overline{X}\leq 74.4)[/tex] = [tex]P(\frac{\overline{X} - \nu }{\frac{\sigma }{\sqrt{n}}}\leq \frac{74.4- 79}{\frac{18}{\sqrt{81}}})[/tex]
= [tex]P(Z\leq -2.3)[/tex]
= .0107
(d) What is Upper P (77.6 less than [tex]\overline{X}[/tex] less than 83.2 ) =
[tex]P(77.6< \overline{X}< 83.2)[/tex] = [tex]P(\frac{77.6- 79}{\frac{18}{\sqrt{81}}})< P(\frac{\overline{X} - \nu }{\frac{\sigma }{\sqrt{n}}}\leq \frac{83.2- 79}{\frac{18}{\sqrt{81}}})[/tex]
= [tex]P(- 0.7< Z< 2.1)[/tex]
= [tex](Z< 2.1) - (Z< -0.7)[/tex]
= 0.9821 - .2420
= 0.7401
