In: Statistics and Probability
A consumer electronics shop is studying the volume of sales of its smart TVs. Historical data from Jan–Dec 2019 show that on average 70 smart TVs were sold per month in 2019. Recent market research suggests that sales may further increase. The shop wants to assess consumer demand to help future inventory planning. The four months Jan–Apr 2020 have been studied: in these months, the average number of smart TVs sold per month was 75 and the population standard deviation was equal to 5. You are required to perform a hypothesis test at the 0.05 significance level, in order to verify whether there was an improvement in smart TV sales. Answer the following questions, rounding any numerical answers to two decimal places.
(a) State the null hypothesis. [2]
(b) State the alternative hypothesis. [2]
(c) State the type of hypothesis test. [2]
(d) Compute the value of the test statistic. [2]
(e) If the critical value approach is carried out: (i) What is the critical value to be used for the comparison? [2] (ii) Provide a graphical representation of the comparison. [3] (iii) Do you reject the null hypothesis? Answer Yes or No and provide the [3] decision rule. (iv) Provide a short interpretation of the outcome (one sentence at most). [2]
We have sample of four months from Jan-Apr 2020, so we have sample size of 4
n=4
Sample mean
The standard deviation of population is given by
Answer(a): The null hypothesis is there was no improvement in smart TV sales
i.e. H0:
Answer(b): The null hypothesis is there was an improvement in smart TV sales
i.e. H1:
Answer(c): The type of hypothesis test is right tailed t test.
Answer(d):
The test statistic to test the above hypotheses is
The value of test statistic for above test is 2
Answer(e):
(i): The critical value of t at 0.05 level of significance with 3df is 2.35
(ii): The right side area of the vertical line is the critical region of t and the left side of the vertical line is acceptance region.
(iii): we have tcal =2, less than tcritical = 2.35 and it indicates that we do not have enough evidence against the null hypothesis to reject it, so we fail to reject the null hypothesis.
That means we do not reject the null hypothesis
(iv): on the basis of above result and evidences, we can interpret that there was no improvement in smart TV sales.