Find critical alpha or significance level

6 A company has two machines, \(A\) and \(B\), which independently fill small bottles with a liquid. The volumes of liquid per bottle, in suitable units, filled by machines \(A\) and \(B\) are denoted by \(x\) and \(y\) respectively. A scientist at the company takes a random sample of 40 bottles filled by machine \(A\) and a random sample of 50 bottles filled by machine \(B\). The results are summarised as follows. $$\sum x = 1120 \quad \sum x ^ { 2 } = 31400 \quad \sum y = 1370 \quad \sum y ^ { 2 } = 37600$$ The population means of the volumes of liquid in the bottles filled by machines \(A\) and \(B\) are denoted by \(\mu _ { A }\) and \(\mu _ { B }\).

1. Test at the \(2 \%\) significance level whether there is any difference between \(\mu _ { A }\) and \(\mu _ { B }\).
2. Find the set of values of \(\alpha\) for which there would be evidence at the \(\alpha \%\) significance level that \(\mu _ { \mathrm { A } } - \mu _ { \mathrm { B } }\) is greater than 0.25.
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5 Dev owns a small company which produces bottles of juice. He uses two machines, \(X\) and \(Y\), to fill empty bottles with juice. Dev is investigating the volumes of juice in the bottles. He chooses a random sample of 35 bottles filled by machine \(X\) and a random sample of 60 bottles filled by machine \(Y\). The volumes of juice, \(x\) and \(y\) respectively, measured in suitable units, are summarised by $$\sum x = 30.8 , \quad \sum x ^ { 2 } = 29.0 , \quad \sum y = 62.4 , \quad \sum y ^ { 2 } = 76.8 .$$ Dev claims that the mean volume of juice in bottles filled by machine \(Y\) is greater than the mean volume of juice in bottles filled by machine \(X\). A test at the \(\alpha \%\) significance level suggests that there is sufficient evidence to support Dev's claim. Find the set of possible values of \(\alpha\). \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-10_2717_33_109_2014} \includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-11_2726_35_97_20}

8 Two machines, \(A\) and \(B\), produce metal rods. Machine \(B\) is new and it is required that its accuracy should be checked against that of machine \(A\). The observed variable is the length of a rod. Random samples of rods, 40 from machine \(A\) and 50 from machine \(B\), are taken and their lengths, \(x _ { A } \mathrm {~cm}\) and \(x _ { B } \mathrm {~cm}\), are measured. The results are summarised by $$\Sigma x _ { A } = 136.48 , \quad \Sigma x _ { B } = 176.35 , \quad \Sigma x _ { B } ^ { 2 } = 630.1940 .$$ The variance of the length of the rods produced by machine \(A\) is known to be \(0.0490 \mathrm {~cm} ^ { 2 }\). The mean lengths of the rods produced by the machines are denoted by \(\mu _ { A } \mathrm {~cm}\) and \(\mu _ { B } \mathrm {~cm}\) respectively.

1. Test, at the \(5 \%\) significance level, the hypothesis \(\mu _ { B } > \mu _ { A }\).
2. Find the set of values of \(a\) for which the null hypothesis \(\mu _ { B } - \mu _ { A } = 0.025\) would not be rejected in favour of the alternative hypothesis \(\mu _ { B } - \mu _ { A } > 0.025\) at the \(a \%\) significance level.
3. For the test in part (i) to be valid,
   1. state whether it is necessary to assume that the two population variances are equal,
   2. state, giving a reason, whether it is necessary to assume that the lengths of rods are normally distributed.

The times taken, in hours, by cyclists from two different clubs, \(A\) and \(B\), to complete a 50 km time trial are being compared. The times taken by a cyclist from club \(A\) and by a cyclist from club \(B\) are denoted by \(t _ { A }\) and \(t _ { B }\) respectively. A random sample of 50 cyclists from \(A\) and a random sample of 60 cyclists from \(B\) give the following summarised data. $$\Sigma t _ { A } = 102.0 \quad \Sigma t _ { A } ^ { 2 } = 215.18 \quad \Sigma t _ { B } = 129.0 \quad \Sigma t _ { B } ^ { 2 } = 282.3$$ Using a 5\% significance level, test whether, on average, cyclists from club \(A\) take less time to complete the time trial than cyclists from club \(B\). A test at the \(\alpha \%\) significance level shows that there is evidence that the population mean time for cyclists from club \(B\) exceeds the population mean time for cyclists from club \(A\) by more than 0.05 hours. Find the set of possible values of \(\alpha\).

A company produces packets of sweets. Two different machines, \(A\) and \(B\), are used to fill the packets. The manager decides to assess the performance of the two machines. He selects a random sample of 50 packets filled by machine \(A\) and a random sample of 60 packets filled by machine \(B\). The masses of sweets, \(x \mathrm {~kg}\), in packets filled by machine \(A\) and the masses of sweets, \(y \mathrm {~kg}\), in packets filled by machine \(B\) are summarised as follows. $$\Sigma x = 22.4 \quad \Sigma x ^ { 2 } = 10.1 \quad \Sigma y = 28.8 \quad \Sigma y ^ { 2 } = 16.3$$ A test at the \(\alpha \%\) significance level provides evidence that the mean mass of sweets in packets filled by machine \(A\) is less than the mean mass of sweets in packets filled by machine \(B\). Find the set of possible values of \(\alpha\).
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A perfume manufacturer had one bottle-filling machine, but because of increased sales a second machine was obtained. In order to compare the performance of the two machines, a random sample of 50 bottles filled by the first machine and a random sample of 60 bottles filled by the second machine were checked. The volumes of the contents from the first machine, \(x _ { 1 } \mathrm { ml }\), and from the second machine, \(x _ { 2 } \mathrm { ml }\), are summarised by $$\Sigma x _ { 1 } = 1492.0 , \quad \Sigma x _ { 1 } ^ { 2 } = 44529.52 , \quad \Sigma x _ { 2 } = 1803.6 , \quad \Sigma x _ { 2 } ^ { 2 } = 54220.84 .$$ The volumes have distributions with means \(\mu _ { 1 } \mathrm { ml }\) and \(\mu _ { 2 } \mathrm { ml }\) for the first and second machines respectively. Test, at the \(2 \%\) significance level, whether \(\mu _ { 2 }\) is greater than \(\mu _ { 1 }\). Find the set of values of \(\alpha\) for which there would be evidence at the \(\alpha \%\) significance level that \(\mu _ { 2 } - \mu _ { 1 } > 0.1\).

A factory produces bottles of spring water. The manager decides to assess the performance of the two machines that are used to fill the bottles with water. He selects a random sample of 60 bottles filled by the first machine \(X\) and a random sample of 80 bottles filled by the second machine \(Y\). The volumes of water, \(x\) and \(y\), measured in appropriate units, are summarised as follows. $$\Sigma x = 58.2 \quad \Sigma x ^ { 2 } = 85.8 \quad \Sigma y = 97.6 \quad \Sigma y ^ { 2 } = 188.6$$ A test at the \(\alpha \%\) significance level shows that the mean volume of water in bottles filled by machine \(X\) is less than the mean volume of water in bottles filled by machine \(Y\). Find the set of possible values of \(\alpha\).

6 A random sample of 50 observations of a random variable \(X\) and a random sample of 60 observations of a random variable \(Y\) are taken. The results for the sample means, \(\bar { x }\) and \(\bar { y }\), and the unbiased estimates for the population variances, \(s _ { x } ^ { 2 }\) and \(s _ { y } ^ { 2 }\), respectively, are as follows. $$\bar { x } = 25.4 \quad \bar { y } = 23.6 \quad s _ { x } ^ { 2 } = 23.2 \quad s _ { y } ^ { 2 } = 27.8$$ A test, at the \(\alpha \%\) significance level, of the null hypothesis that the population means of \(X\) and \(Y\) are equal against the alternative hypothesis that they are not equal is carried out. Given that the null hypothesis is not rejected, find the set of possible values of \(\alpha\).

6 The random variable \(X\) was assumed to have a normal distribution with mean \(\mu\). Using a random sample of size 128, a significance test was carried out using the following hypotheses. \(\mathrm { H } _ { 0 } : \mu = 30\) \(\mathrm { H } _ { 1 } : \mu > 30\) It was found that \(\sum x = 3929.6\) and \(\sum x ^ { 2 } = 123483.52\). The conclusion of the test was to reject the null hypothesis.

1. Determine the range of possible values of the significance level of the test.
2. It was subsequently found that \(X\) was not normally distributed. Explain whether this invalidates the conclusion of the test.

A random sample of 50 observations of a random variable \(X\) and a random sample of 60 observations of a random variable \(Y\) are taken. The results for the sample means, \(\bar{x}\) and \(\bar{y}\), and the unbiased estimates for the population variances, \(s_x^2\) and \(s_y^2\), respectively, are as follows. $$\bar{x} = 25.4 \quad \bar{y} = 23.6 \quad s_x^2 = 23.2 \quad s_y^2 = 27.8$$ A test, at the \(\alpha\%\) significance level, of the null hypothesis that the population means of \(X\) and \(Y\) are equal against the alternative hypothesis that they are not equal is carried out. Given that the null hypothesis is not rejected, find the set of possible values of \(\alpha\). [5]