Respuesta :
Answer:
See below
Step-by-step explanation:
Rayovac type-D batteries
Mean
[tex] \bf \bar x_1[/tex] = 21 h
Standard deviation
[tex] \bf s_1[/tex] = 1.38 h
Find the probability (to four places after the decimal) that a single Rayovac battery lasts under 22 hours:
Here we use the Normal N(0,1)
[tex] \bf P(z<\frac{22-21}{1.38})=P(z<0.7246)=0.7657[/tex]
To find this value use either a table or a spreadsheet.
In Excel
NORMDIST(0.7246,0,1,1)
In OpenOffice Calc
NORMDIST(0.7246;0;1;1)
Whenever we have a sample size less than 30, we will be using the Student's t distribution with (sample size -1) degrees of freedom instead of the Normal.
Find the probability that the mean "continuous use-life" of 4 randomly selected Rayovac batteries is less than 22 hours:
[tex] \bf P(t<\frac{22-21}{1.38/\sqrt{4}})=P(t<1.4493)=0.8784[/tex]
To find this value use either a table or a spreadsheet.
In Excel
1-TDIST(1.4493,3,1)
In OpenOffice Calc
1-TDIST(1.4493;3;1)
Find the probability that the mean "continuous use-life" of 16 randomly selected Rayovac batteries is less than 22 hours:
[tex] \bf P(t<\frac{22-21}{1.38/\sqrt{16}})=P(t<2.8986)=0.9945[/tex]
To find this value use either a table or a spreadsheet.
In Excel
1-TDIST(2.8986,15,1)
In OpenOffice Calc
1-TDIST(2.8986 ;15;1)
Find the probability that the mean "continuous use-life" of 64 randomly selected Rayovac batteries is less than 22 hours:
Since the sample size is greater than 30 we can use back the Normal
[tex] \bf P(z<\frac{22-21}{1.38/\sqrt{64}})=P(z<5.7971)\approx 1[/tex]
To find this value use either a table or a spreadsheet.
In Excel
NORMDIST(5.7971,0,1,1)
In OpenOffice Calc
NORMDIST(5.7971;0;1;1)
Duracell type-D batteries
Mean
[tex] \bf \bar x_2[/tex] = 24 h
Standard deviation
[tex] \bf s_2[/tex] = 2.13 h
Find the probability that a single Duracell battery lasts under 22 hours
Here we use the Normal N(0,1)
[tex] \bf P(z<\frac{22-24}{2.13})=P(z<-0.9390)=0.1739[/tex]
To find this value use either a table or a spreadsheet.
In Excel
NORMDIST(-0.9390,0,1,1)
In OpenOffice Calc
NORMDIST(-0.9390;0;1;1)
Find the probability that the mean "continuous use-life" of 4 randomly selected Duracell batteries is less than 22 hours:
[tex] \bf P(t<\frac{22-24}{2.13/\sqrt{4}})=P(t< -1.8779)=0.0785[/tex]
To find this value use either a table or a spreadsheet.
In Excel
TDIST(1.8779,3,1)
In OpenOffice Calc
TDIST(1.8779;3;1)
Find the probability that the mean "continuous use-life" of 9 randomly selected Duracell batteries is less than 22 hours:
[tex] \bf P(t<\frac{22-24}{2.13/\sqrt{9}})=P(t<- 2.8169)=0.0113[/tex]
To find this value use either a table or a spreadsheet.
In Excel
TDIST(2.8169,8,1)
In OpenOffice Calc
TDIST(2.8169;8;1)
Find the probability that the mean "continuous use-life" of 25 randomly selected Duracell batteries is less than 22 hours:
[tex] \bf P(t<\frac{22-24}{2.13/\sqrt{25}})=P(t<- 4.6948)\approx 0[/tex]
To find this value use either a table or a spreadsheet.
In Excel
TDIST(4.6948,24,1)
In OpenOffice Calc
TDIST(4.6948;24;1)
Find the probability that a randomly selected Rayovac battery lasts longer than a randomly selected Duracell battery given random samples of four of each kind.
[tex] \bf P(t>\frac{\bar x_1-\bar x_2}{\sqrt{s_1^2/4+s_2^2/4}})=P(t>\frac{21-24}{\sqrt{(1.38)^2/4+(2.13)^2/4}})=\\=P(t>-2.364)=1-P(t<-2.364)=1-0.0496=0.9504[/tex]
