A property and casualty insurance company (which provides fire coverage for dwel

Question

A property and casualty insurance company (which provides fire coverage for dwellings) felt that the mean distance from a home to the nearest fire department in rural Alabama was at least 10 miles. It set its fire insurance rates accordingly. Members of the community set out to show that the mean distance was less than 10 miles due to the increased number of volunteer fire departments. This, they felt, would convince the insurance company to lower its rates. They randomly identify 64 homes and measure the distance to the nearest fire department for each. The resulting sample mean was 8.7 miles. If = 3.5 miles, does the sample show sufficient evidence to support the community’s claim? Use the four step process for Hypothesis Testing. (8 points)

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An oil additive for automobiles claims that it increases fuel mileage by at least a mean of 2.3 mpg with a standard deviation of .5 mpg. A group of AP Statistics students decide to test this claim. They randomly select 49 people to test the oil additive and found that drivers using the additive increased their fuel mileage by a mean of 2.1 mpg, with a standard deviation of .5. Is this sufficient evidence to reject the null hypothesis in favor of the manufacturer’s claim? Use = 0.01 Use the four step process for Hypothesis Testing. (8 points)

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Explanation / Answer

Q1.

Given that,
population mean(u)=10
standard deviation, =3.5
sample mean, x =8.7
number (n)=64
null, Ho: >=10
alternate, H1: <10
level of significance, = 0.05
from standard normal table,left tailed z /2 =1.645
since our test is left-tailed
reject Ho, if zo < -1.645
we use test statistic (z) = x-u/(s.d/sqrt(n))
zo = 8.7-10/(3.5/sqrt(64)
zo = -2.97143
| zo | = 2.97143
critical value
the value of |z | at los 5% is 1.645
we got |zo| =2.97143 & | z | = 1.645
make decision
hence value of | zo | > | z | and here we reject Ho
p-value : left tail - ha : ( p < -2.97143 ) = 0.00148
hence value of p0.05 > 0.00148, here we reject Ho
ANSWERS
---------------
null, Ho: >=10
alternate, H1: <10
test statistic: -2.97143
critical value: -1.645
decision: reject Ho
p-value: 0.00148
we don't have evidence to support the mean distance from a home to the nearest
fire department in rural Alabama was at least 10 miles

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