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help me with statistics!!!?

A manufacturer of small appliances employs a market research firm to estimate retail sales of its products by gathering information from a sample of retail stores. This month an SRS of 75 stores in the Midwest sales region finds that these stores sold an average of 24 of the manufacturer’s hand mixers, with standard deviation 11 (of the sample).

a)The distribution of sales is strongly skewed right, because there are many small stores and few very large stores. The use of the t-model is reasonably safe despite the violation of the normality assumption. Why?

b)Give the 95% confidence interval for the mean number of mixers sold by all store in the region.

c)A supervisor of the manufacturing company believes the average sale of hand mixers in the region is about 30. He believes the marketing firm is underestimating the average sales. Test the supervisors claim at a 5% level of significance.

a) the central limit theorem states that if we have a sample X1, X2, X3, … , Xn from any population with mean μ and standard deviation σ, then the mean of the sample, if n, the sample size is large enough, will be approximately normally distributed. With a sample size of 75 it is reasonable to assume the mean will be normally distributed and thus the t test will suffice. However, if you have the original data set it would be advisable to verify the data is normally distributed to confirm the assumption by using a probability plot.

b)

before anything else, for small sample CI’s you must have normal data.

small sample confidence intervals are used to find a region in which we are 100 (1-α)% confident the true value of the parameter is in the interval.

For small sample confidence intervals about the mean you have:

xBar ± t * sx / sqrt(n)

where xBar is the sample mean
t is the t score for having α% of the data in the tails, i.e., P( T > |t|) = α
note that the student t has n – 1 degrees of freedom
sx is the sample standard deviation
n is the sample size

the t statistic here is:

P( |t_74| > t) = 0.05
P( t_74 > t ) = 0.025
t = 1.992543

the CI is:

24 ± 1.992543 * 11 / sqrt(75)

= (21.46913, 26.53087)

c)

you don’t need to conduct a hypothesis test to find this answer. The value of 30 is not contained within the 95% CI which tells us that at the 5% significance level we can reject the hypothesis that the average number of sales is 30. But to show the process of a hypothesis test I will go through it here.

Hypothesis Test for mean:

assuming you have a large enough sample such that the central limit theorem holds and the mean is normally distributed then to test the null hypothesis H0: μ = Δ

find the test statistic z = (xBar – Δ) / (sx / sqrt(n))

where xbar is the sample average
sx is the sample standard deviation
n is the sample size

The p-value of the test is the area under the normal curve that is in agreement with the alternate hypothesis.

H1: μ > Δ; p-value is the area to the right of z
H1: μ < Δ; p-value is the area to the left of z
H1: μ ≠ Δ; p-value is the area in the tails greater than |z|

for a small sample test for the mean every thing is the same save the test statistic is a t statistic with n - 1 degrees of freedom. However, in this case the underlying distribution must be normal for this test to be valid.

You have:

H0: μ = 30 vs H1: μ ≠ 30

the test statistic is t = (24 - 30) / ( 11/sqrt(75) ) = 4.723775

the p-value is:

P( |t_74| > 4.723775)
= 2 * P( t_74 < -4.723775)
= 2 * 5.372446e-06
= 0.00001074489

since the p-value is less than the significance level we reject the null hypothesis and conclude the mean is different from 30.

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