Question

In a test of the hypothesis that the population mean is smaller than 50, a random sample of 10 observations is selected from the population and has a mean of 47.0 and a standard deviation of 4.1. Assume this population is normal.

a) Set up the two hypotheses for this test. Make sure you write them properly.

b) Check the assumptions that need to hold to perform this hypothesis test.

c) Calculate the t-statistic associated with the sample.

d) Graphically interpret the p-value for this test, that is, i) draw a (nice) graph with a t-distribution (remember of the number of degrees of freedom) ii) locate on the graph the t-statistic you found in part (c) iii) mark the P-value on the graph

e) Calculate the P-value for this test.

f) Statistically interpret the P-value for this test.

g) Let the level of significance α = 2.5%. Using P-value, make a conclusion for your test (write a complete sentence for full credit).

h) Let the level of significance α = 2.5%. Find the related critical value tα.

i) What is the rejection region (RR) implied by α = 2.5% ?

j) Draw the RR on your graph on page 1, part (d).

k) Using the RR, make a conclusion for your test (write a complete sentence for full credit).

Answer 1

Part a

Null hypothesis: H₀: The population mean is 50.

Alternative hypothesis: H_a: The population mean is less than 50.

H₀: µ = 50 versus H_a: µ < 50

(one tailed /left tailed test)

Part b

We already assume that population is normal from where sample is taken.

Part c

Test statistic is given as below:

t = (Xbar - µ)/[S/sqrt(n)]

We are given Xbar = 47, S=4.1, n=10, df=n -1 = 9, α = 0.025,

Critical t value = -2.2622 (by using t-table)

t = (47 – 50)/[4.1/sqrt(10)]

t = -2.3139

Part d

Required graph is given as below:

Part e

P-value = 0.0230

(by using t-table)

Part f

P-value is the significance level at which we null hypothesis is rejected. For this scenario, the p-value is the area of t < -2.3139.

Part g

We are given α = 0.025

P-value = 0.0230

P-value < α = 0.025

So, we reject the null hypothesis H₀

There is sufficient evidence to conclude that population mean is less than 50.

Part h

Critical value t_alpha = -2.2622

Part i

Reject H₀ when t<-2.2622

(by using t-table)

Part j

Please see the above fig.

Part k

We have t = -2.3139 < t_alpha = -2.2622

So, we reject the null hypothesis

There is sufficient evidence to conclude that population mean is less than 50.