Respuesta :
Answer:
(1) The probability that the sample mean is Greater than 59 = 0.778
(2) The probability that the sample mean is less than 59 = .3632
(3) The probability that the sample mean is is Between 54 and 59 = 0.559
Step-by-step explanation:
Given -
Mean [tex](\nu )[/tex] = 55
Standard deviation [tex](\sigma )[/tex] = 14
Sample size ( n ) = 25
the probability that the sample mean [tex](\overline{X})[/tex] is Greater than 59 =
[tex]P(\overline{X} > 59 )[/tex] = [tex]P(\frac{\overline{X} - \nu}{\frac{\sigma}{\sqrt{n}}} > \frac{59 - 55 }{\frac{14}{\sqrt{25}}})[/tex]
= [tex]P(Z > 1.428 )[/tex]
= 1 - [tex]P(Z < 1.428 )[/tex]
= 1 - 0.9222 = .0778
the probability that the sample mean [tex](\overline{X})[/tex] is Less than 54 =
[tex]P(\overline{X}< 54 )[/tex] = [tex]P(\frac{\overline{X} - \nu}{\frac{\sigma}{\sqrt{n}}} < \frac{54 - 55 }{\frac{14}{\sqrt{25}}})[/tex]
= [tex]P(Z< -.357 )[/tex]
= .3632
the probability that the sample mean is Between 54 and 59 =
[tex]P(54< \overline{X}< 59)[/tex] = [tex]P(\frac{54 - 55 }{\frac{14}{\sqrt{25}}}< \frac{\overline{X} - \nu}{\frac{\sigma}{\sqrt{n}}} < \frac{59 - 55 }{\frac{14}{\sqrt{25}}})[/tex]
= [tex]P(-.357< Z< 1.428)[/tex]
= [tex]P(Z< 1.428 ) - P(< -.357)[/tex]
= .9222 - .3632
= 0.559
