A particular model of power lawn mower yields a mean of 26 hours of use per gall

Question

A particular model of power lawn mower yields a mean of 26 hours of use per gallon of gasoline according to company test.A new motor design for the mower has been developed and the designers claim mean hours of use with the mean motor exceed 26 hours per gallon management wishes to test this claim against the alternative that mean hours of use with the new motor does not exceed 26 hour per gallon.if it does exceed 26 hours per gallon they plan to adopt the new motor for this mover
1_ specify the alternatives and associated management action for this statistical test .
2_ suppose that management has a random sample of n= 49 mowers with the new motor design built and test.the following decision rule was specified
if Xbar less than or equal to 26.3 conclude HA=u less than or equal to 26.0
if Xbar greater than 26.3 conclude Hb= u=26.0
determine the probability of concluding HA and Hb if the mean hours of use per gallon with the new motor is actually 25.9.it can be assumed from past experience s= 1.4
construct the power curve and error curve for this decision rule.use population standard deviation to be 1.4 and ordinates corresponding to u=25.6, 26.1, 26.3, 26.5 26.7 while constructing your curves
3_using your result above sketch power and error curve and interpretation both kind of risks

Explanation / Answer

Given that,
population mean(u)=26
standard deviation, =1.4
sample mean, x =26.3
number (n)=49
null, Ho: <26
alternate, H1: >26
level of significance, = 0.05
from standard normal table,right tailed z /2 =1.645
since our test is right-tailed
reject Ho, if zo > 1.645
we use test statistic (z) = x-u/(s.d/sqrt(n))
zo = 26.3-26/(1.4/sqrt(49)
zo = 1.5
| zo | = 1.5
critical value
the value of |z | at los 5% is 1.645
we got |zo| =1.5 & | z | = 1.645
make decision
hence value of |zo | < | z | and here we do not reject Ho
p-value : right tail - ha : ( p > 1.5 ) = 0.06681
hence value of p0.05 < 0.06681, here we do not reject Ho
ANSWERS
---------------
null, Ho: <26
alternate, H1: >26
test statistic: 1.5
critical value: 1.645
decision: do not reject Ho
p-value: 0.06681

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