Two-sample t-test with summary statistics

3. It is known from past evidence that the weight of coffee dispensed into jars by machine \(A\) is normally distributed with mean \(\mu _ { \mathrm { A } }\) and standard deviation 2.5 g . Machine \(B\) is known to dispense the same nominal weight of coffee into jars with mean \(\mu _ { B }\) and standard deviation 2.3 g . A random sample of 10 jars filled by machine \(A\) contained a mean weight of 249 g of coffee. A random sample of 15 jars filled by machine \(B\) contained a mean weight of 251 g .

1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the population mean weight dispensed by machine B is greater than that of machine A .
2. Write down an assumption needed to carry out this test.

6

1. The independent random variables \(S\) and \(L\) have means \(\mu _ { S }\) and \(\mu _ { L }\) respectively, and a common variance of \(\sigma ^ { 2 }\). The variable \(\bar { S }\) denotes the mean of a random sample of \(n\) observations on \(S\) and the variable \(\bar { L }\) denotes the mean of a random sample of \(n\) observations on \(L\). Find a simplified expression, in terms of \(\sigma ^ { 2 }\), for the variance of \(\bar { L } - 2 \bar { S }\).
2. A machine fills both small bottles and large bottles with shower gel. It is known that the volume of shower gel delivered by the machine is normally distributed with a standard deviation of 8 ml .
   1. A random sample of 25 small bottles filled by the machine contained a mean volume of \(\bar { s } = 258 \mathrm { ml }\) of shower gel. An independent random sample of 25 large bottles filled by the machine contained a mean volume of \(\bar { l } = 522 \mathrm { ml }\) of shower gel. Investigate, at the \(10 \%\) level of significance, the hypothesis that the mean volume of shower gel in a large bottle is more than twice that in a small bottle.
      [0pt] [7 marks]
   2. Deduce that, for the test of the hypothesis in part (b)(i), the critical value of \(\bar { L } - 2 \bar { S }\) is 4.585 , correct to three decimal places.
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   3. In fact, the mean volume of shower gel in a large bottle exceeds twice that in a small bottle by 10 ml . Determine the probability of a Type II error for a test of the hypothesis in part (b)(i) at the 10\% level of significance, based upon random samples of 25 small bottles and 25 large bottles.
      [0pt] [4 marks]

7. Branwen intends to buy a new bike, either a Cannotrek or a Bianchondale. If there is evidence that the difference in the mean times on the two bikes over a 10 km time trial is more than 1.25 minutes, she will buy the faster bike. Otherwise, she will base her decision on other factors. She negotiates a test period to try both bikes. The times, in minutes, taken by Branwen to complete a 10 km time trial on the Cannotrek may be modelled by a normal distribution with mean \(\mu _ { C }\) and standard deviation \(0 \cdot 75\). The times, in minutes, taken by Branwen to complete a 10 km time trial on the Bianchondale may be modelled by a normal distribution with mean \(\mu _ { B }\) and standard deviation \(0 \cdot 6\). During the test period, she completes 6 time trials with a mean time of 19.5 minutes on the Cannotrek, and 5 time trials with a mean time of 17.3 minutes on the Bianchondale. She calculates a \(p \%\) confidence interval for \(\mu _ { C } - \mu _ { B }\).

1. What would be the largest value of \(p\) that would lead Branwen to base her purchasing decision on the time trials, without considering other factors?
2. State an assumption you have made in part (a).

1. A company produces two colours of candles, blue and white. The standard deviation of the burning times of the blue candles is 2.6 minutes and the standard deviation of the burning times of the white candles is 2.4 minutes.

Nissim claims that the mean burning time of blue candles is more than 5 minutes greater than the mean burning time of white candles. A random sample of 90 blue candles is found to have a mean burning time of 39.5 minutes. A random sample of 80 white candles is found to have a mean burning time of 33.7 minutes.

1. Stating your hypotheses clearly, use a suitable test to assess Nissim's belief. Use a \(1 \%\) level of significance.
2. Explain how the hypothesis test in part (a) would be carried out differently if the variances of the burning times of candles were unknown. The burning times for the candles may not follow a normal distribution.
3. Describe the effect this would have on the calculations in the hypothesis test in part (a). Give a reason for your answer.

The weekly salaries of employees at two large electronics companies, \(A\) and \(B\), are being compared. The weekly salary of an employee from company \(A\) and an employee from company \(B\) are denoted by \(\\)x\( and \)\\(y\) respectively. A random sample of 50 employees from company \(A\) and a random sample of 40 employees from company \(B\) give the following summarised data. $$\Sigma x = 5120 \quad \Sigma x^2 = 531000 \quad \Sigma y = 3760 \quad \Sigma y^2 = 375135$$

1. The population mean salaries of employees from companies \(A\) and \(B\) are denoted by \(\\)\mu_A\( and \)\\(\mu_B\) respectively. Using a 5\% significance level, test the null hypothesis \(\mu_A = \mu_B\) against the alternative hypothesis \(\mu_A \neq \mu_B\). [8]
2. State, with a reason, whether any assumptions about the distributions of employees' salaries are needed for the test in part (i). [1]

A random sample of 100 classical CDs produced by a record company had a mean playing time of 70.6 minutes and a standard deviation of 9.1 minutes. An independent random sample of 80 CDs produced by a different company had a mean playing time of 67.2 minutes with a standard deviation of 8.4 minutes.

1. Using a 1\% level of significance, test whether or not there is a difference in the mean playing times of the CDs produced by these two companies. State your hypotheses clearly. [8]
2. State an assumption you made in carrying out the test in part (a). [1]

A random sample of 100 classical CDs produced by a record company had a mean playing time of 70.6 minutes and a standard deviation of 9.1 minutes. An independent random sample of 120 CDs produced by a different company had a mean playing time of 67.2 minutes with a standard deviation of 8.4 minutes.

1. Using a 1\% level of significance, test whether or not there is a difference in the mean playing times of the CDs produced by these two companies. State your hypotheses clearly. [8]
2. State an assumption you made in carrying out the test in part (a). [1]

A sociologist was studying the smoking habits of adults. A random sample of 300 adult smokers from a low income group and an independent random sample of 400 adult smokers from a high income group were asked what their weekly expenditure on tobacco was. The results are summarised below.

1. Using a 5\% significance level, test whether or not the two groups differ in the mean amounts spent on tobacco. [9]
2. Explain briefly the importance of the central limit theorem in this example. [2]