contestada

A confidence interval (CI) is desired for the true average stray-load loss u (watts) for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that strayload loss is normally distributed with o = 3.0. (1). Construct a 95% CI for u when n = 25 and x = 58.3. (10 points) (2). Construct a 95% CI for u when n = 100 and 1 = 58.3. (10 points) (3). Construct a 99% CI for u when n = 100 and x = 58.3. (10 points) (4). How large must n be if the half-width of the 99% interval for u is to be 0.5? (10 points)

Respuesta :

Answer:

A) CI = (57.12 , 59.48)

B) CI = (57.71 , 58.89)

C) CI = (57.53 , 59.07)

D) n = 239.63

Step-by-step explanation:

a)

given data:

mean, [tex]\bar X = 58.3[/tex]

standard deviation, σ = 3

sample size, n = 25Given CI level is 95%, hence α = 1 - 0.95 = 0.05

α/2 = 0.05/2 = 0.025,

Zc = Z(α/2) = 1.96

[tex]ME = Zc * σ \sqrt{n}[/tex]

[tex]ME = 1.96 * 3 \sqrt{25}[/tex]

ME = 1.18

[tex]CI = (\bar X - Zc * s\sqrt{n}  , \barX + Zc * s\sqrt{n})[/tex]

[tex]CI = (58.3 - 1.96 * 3\sqrt{25} , 58.3 + 1.96 * 3\sqrt{25})[/tex]

CI = (57.12 , 59.48)

b)

Given data:

mean, [tex]\bar X = 58.3[/tex]

standard deviation, σ = 3

sample size, n = 100

Given CI level is 95%, hence α = 1 - 0.95 = 0.05

α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

[tex]ME = zc * σ \sqrt{n}  [/tex]

[tex]ME = 1.96 * 3\sqrt{100}  [/tex]

ME = 0.59

[tex]CI = (\bar X - Zc * s\sqrt{n}  , \barX + Zc * s\sqrt{n})[/tex]

[tex]CI = (58.3 - 1.96 * 3\sqrt{100} , 58.3 + 1.96 * 3\sqrt{100})[/tex]

CI = (57.71 , 58.89)

c)

sample mean, [tex]\bar X = 58.3[/tex]

sample standard deviation, σ = 3

sample size, n = 100

Given CI level is 99%, hence α = 1 - 0.99 = 0.01

α/2 = 0.01/2 = 0.005, Zc = Z(α/2) = 2.58

[tex]ME = Zc * σ \sqrt{n}  [/tex]

[tex]ME = 2.58 * 3\sqrt{100}[/tex]

ME = 0.77

[tex]CI = (\bar X - Zc * s\sqrt{n}  , \barX + Zc * s\sqrt{n})[/tex]

[tex]CI = (58.3 - 2.58 * 3\sqrt{100} , 58.3 + 2.58 * 3/\sqrt{100}[/tex]

CI = (57.53 , 59.07)

D)

Given data:

Significance Level, α = 0.01,

Margin or Error, E = 0.5,

σ = 3

The critical value for α = 0.01 is 2.58.

for calculating population mean we used

[tex]n \geq (zc *σ/E)^2[/tex]

[tex]n = (2.58 * 3/0.5)^2[/tex]

n = 239.63

Otras preguntas

Q&A Education