Worksheet 5

Q1: For this question, use the petrol consumption data from Worksheet 4, (included the data for worksheet 4)

8.9 10.4 6.3 8.9 10.8 10.8 8.2

8.1 6.6 10.1 6.3 8.3 10.1 6.6

8.1 6.1 7.6 10.4 8.2 10.3

Question 1. At the 5% level of significance, test the null hypothesis that the population mean equals 9.000 against the alternative that the population mean is less than 9.000, by following

the steps below.

(a) Find the value of x¯ for this test.

**To get any marks for question 1, you must submit a correct answer to (a).**

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(b) Find the value of s / Squrt (n) for this test.

(c) Give the hypothesized mean μ for this test.

Give your answers to (a){(c) to **4 significant figures**.

(d) From the answers to (a){(c), find Tobserved .

To get marks for (d), your answer must be consistent

with your answers to (a){(c).

(e) Use Minitab to find the critical value, c, for thistest. Your answer must be a negative number. Please give your answer to **4 decimal places**.

Are the following statements true?

(f) Tobserved < c

(g) As a result of the test, we reject the null hypothesis.

(h) The conclusion of the test is that the population mean is equal to 9.000.

To get marks for (f), (g) and (h), your answers must be consistent with your answers to (d) and (e).

Q2: A new random sample of petrol consumption data is obtained. As before, the null hypothesis is H0: μ = 9.000 against the alternative H1: μ < 9.000, and the 5% level of significance is used. The sample size is n = 13. The observed value of the T statistic

is found to be Tobserved = -1.423.

(a) Use Minitab to find the P-value for this test. Give your answer to **3 decimal places**.

Are the following statements true?

(b) The P-value is greater than the level of significance.

(c) As a result of the test, we accept the null hypothesis.

(d)We conclude that the population mean is less than 9.000.

To get marks for (b), (c) and (d), your answers must be consistent with your answer to (a).

Q3: A cosmetics company sells on average 5650 bottles of a fragrance per week. They change the package design, and find that the sales in the ten weeks after

the change are 6640, 6570, 4810, 7180, 6480, 6220, 7000, 6580, 6630, 5940. The company wants to know if the package design has increased sales.

(a) Which of the following statements is the null hypothesis for this test?

(i) H0: μ > 5650

(ii) H0: μ ≠ 5650

(iii) H0: μ < 5650

(iv) H0: μ = 5650

(b) Which of the following statements is the alternative hypothesis for this test?

(i) H1: μ > 5650

(ii) H1: μ ≠ 5650

(iii) H1: μ < 5650

(iv) H1: μ = 5650

(c) Use Minitab to find the observed value of the T statistic for this test. Please give your answer to **2 decimal places**.

(d) Use Minitab and assume a 5% level of significance to find the critical value, c, for this test. Please give your answer to **4 decimal places**.

(e) At the 5% level of significance, do you reject the null hypothesis?

(f) At the 5% level of significance, is there evidence that the package design has increased sales?

Q4: An insurance company knows that the average amount by which houses are underinsured in Leeds is $33000. They want to find out about the average amount for houses in London, so they take a sample of 14 houses in London. Their question is: are houses in

London on average underinsured by a larger amount than those in Leeds?

Here are four statements:

(i) The population mean for London is greater than $33000.

(ii) The population mean for London is different from $33000.

(iii) The population mean for London is less than $33000.

(iv) The population mean for London is $33000.

(a) Which of the above statements is the null hypothesis for this test?

(b) Which of the above statements is the alternative hypothesis for this test?

(c) The observed value of the T statistic for this test is Tobserved = 1:222. Find the corresponding P-value. Give your answer to **3 decimal places**.

(d) At the 5% level of signi_cance, do you accept the null hypothesis?

(e) At the 5% level of signi_cance, is there evidence that houses in London are on average underinsured by a larger amount than those in Leeds?

Q5: A manufacturer produces chocolate. Each batch is meant to contain 60 bars on average. An average above or below this standard is undesirable. A sample of 11 batches is taken and the following numbers of bars are found:

62, 62, 61, 63, 60, 61, 59, 59, 64, 61, 63.

The manufacturer wants to know if there is evidence that the production process is not functioning correctly.

Here are four statements:

(i) The (population) average number of bars in a batch is greater than 60.

(ii) The (population) average number of bars in a batch is di_erent from 60.

(iii) The (population) average number of bars in a batch is less than 60.

(iv) The (population) average number of bars in a batch is 60.

(a) Which of the above statements is the null hypothesis for this test?

(b) Which of the above statements is the alternative hypothesis for this test?

(c) Use Minitab to find the observed value of the T statistic for this test. Please give your answer to **2 decimal places**.

(d) Use Minitab and assume the 5% level of significance to and the critical value, c, for this test. Your answer should be a positive number. Give your answer to **4 decimal places**.

(e) At the 5% level of significance, do you reject the null hypothesis?

(f) At the 5% level of significance, is there evidence that the process is malfunctioning?

**Verify your conclusions in two ways: first by comparing the P-value for the test to the level of significance, then by checking whether the hypothesized mean (=60) lies inside the 95% confidence interval.**

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Q6: The chocolate manufacturer of Q5 above takes a new sample of batches, this time of size n = 14, and performs the same test as before. He finds Tobserved = -1.929.

(a) Use Minitab to determine the P-value for this test. Give your answer to 3 decimal places.

(b) At the 5% level of significance, do you accept the null hypothesis?

(c) At the 5% level of significance, is there evidence that the process is malfunctioning?