In: Statistics and Probability
An insurance company claims that, on average, they save their customers £196. A competitor company is interested in showing that this value is an overestimate. They randomly sample 10 customers who saved 96, 207, 90, 24, 0, 263, 20, 66, 135, 189 pounds each. The variance of the sample is £7755.
(a) What are the assumptions for conducting a hypothesis test around this data? Are these satisfied?
(b) What is your null hypothesis?
(c) What is your alternative hypothesis?
(d) Calculate the p-value for the hypotheses you proposed in (b) & (c).
(e) What is your conclusion when conducting a 95% hypothesis test?
(f) Do you agree with the competitor company that the savings advertised is an overesti- mate? State your reasoning.
(g) Describe an alternative method you could have used to calculate a p-value.
A)
Assumptions are that the sample is randomly chosen
And the population is normally distributed.
B)
Null hypothesis Ho : u = 196
C)
Alternate hypothesis Ha : u < 196
(As the claim is that this value is an overestimate, which means they are conducting a test to test if the actual mean is less than this value)
D)
First we need to estimate the sample mean and sample standard deviation of the given data
Sample mean = 109
S.d = 88.0669
As the population standard deviation is unknown, we will use t test.
Test statistics t = (sample mean - claimed mean)/(s.d/√n)
Claimed mean = 196
N = sample size = 10
t = −3.123967761266
For degrees of freedom n-1, 9 and test statistics of -3.124
P-value is = 0.00612
E)
95% means alpha = 1-0.95 = 0.05
So given significance level is 0.05
As the obtained P-Value is less than the 0.05, we reject the null hypothesis.
F)
Yes, as we rejected null hypothesis, so we have enough evidence to support the claim that u < 196