Question

The mean and standard deviation of the lifetimes of 9 randomly selected Duracell batteries are 42 days and 20 days, respectively. The mean and standard deviation of the lifetimes of 11 randomly selected Eveready batteries are 45 days and 17 days, respectively. Suppose that the lifetime of a Duracell battery has NORM(μ_X,σ^2 ) and the lifetime of an Eveready battery has NORM(μ_Y,σ^2 ).

a) Find an unbiased estimate of σ^2.

b) Calculate an 96% confidence interval for μ_X-μ_Y.

c) Test whether the mean lifetime of all Duracell batteries is less than the mean lifetime of all Eveready batteries. State the null and alternative hypotheses, determine the critical region, make a decision based on α=0.05 and write your conclusion.

Answer 1

for sample 1

n1=9

X1bar=42

sd1=20

Same way for sample 2

n2=11

X2bar=45

sd2=17

b)

For 96% conf level of mean we need to perform the z test

Z critical from the table for 96% conf level is -/+2.014091

Z= [(X1bar-X2bar)-(mu1-mu2)]/sqrt[(sd1^2/n1+sd2^2/n2)]

Solving for mu1-mu2 for two different values of z +/- 2.014091 with the given xbar and sd we get

mu1-mu2 = -19.9372

mu1-mu2= 13.9372

So the 96% conf range is -19.9372 to 13.9372

c) Ho: mu1=mu2

Ha: mu11<mu2 (Hypothesis under test)

So.,

finding t stat we get

t= [(X1bar-X2bar)-(mu1-mu2)]/sqrt[(sd1^2/n1+sd2^2/n2)]

where mu1-mu2=0 (null hypothesis)

So., t = -0.357

While the 95% conf critical t value from t table is -1.734064

looking from t table for the area under the curve as 5% with n1+n2-2 = 18 degrees of freedom

So here t value > t critical

and so we can't reject the Ho

So we can conclude that the mean battery life for everady and Duracell are same only.