F-test and chi-squared for variance

The random variable \(X\) has an \(F\)-distribution with 8 and 12 degrees of freedom. Find P\(\left(\frac{1}{5.67} < X < 2.85\right)\). [4]

The random variable \(X\) has a \(\chi^2\)-distribution with 9 degrees of freedom.

1. Find P(2.088 < \(X\) < 19.023). [3]

The random variable \(Y\) follows an \(F\)-distribution with 12 and 5 degrees of freedom.

1. [(b)] Find the upper and lower 5\% critical values for \(Y\). [3]

(Total 6 marks)

2. The weights, in grams, of apples are assumed to follow a normal distribution. The weights of apples sold by a supermarket have variance \(\sigma _ { s } { } ^ { 2 }\). A random sample of 4 apples from the supermarket had weights $$\text { 114, 100, 119, } 123 .$$

1. Find a 95\% confidence interval for \(\sigma _ { s } ^ { 2 }\). The weights of apples sold on a market stall have variance \(\sigma _ { M } ^ { 2 }\). A second random sample of 7 apples was taken from the market stall. The sample variance \(s _ { M } ^ { 2 }\) of the apples was 318.8.
2. Stating your hypotheses clearly test, at the \(1 \%\) levcel of significnace, whether or not there is evidence that \(\sigma _ { M } ^ { 2 } > \sigma _ { s } ^ { 2 }\).

The lengths, \(x\) mm, of the forewings of a random sample of male and female adult butterflies are measured. The following statistics are obtained from the data. \includegraphics{figure_3}

1. Assuming the lengths of the forewings are normally distributed test, at the 10\% level of significance, whether or not the variances of the two distributions are the same. State your hypotheses clearly. [7]
2. Stating your hypotheses clearly test, at the 5\% level of significance, whether the mean length of the forewings of the female butterflies is less than the mean length of the forewings of the male butterflies. [6]

7. A psychologist gives a test to students from two different schools, \(A\) and \(B\). A group of 9 students is randomly selected from school \(A\) and given instructions on how to do the test.
A group of 7 students is randomly selected from school \(B\) and given the test without the instructions. The table shows the time taken, to the nearest second, to complete the test by the two groups. Stating your hypotheses clearly,

1. test at the \(10 \%\) significance level, whether or not the variance of the times taken to complete the test by students from school \(A\) is the same as the variance of the times taken to complete the test by students from school \(B\). (You may assume that times taken for each school are normally distributed.)
2. test at the \(5 \%\) significance level, whether or not the mean time taken to complete the test by students from school \(A\) is greater than the mean time taken to complete the test by students from school \(B\).
3. Why does the result to part (a) enable you to carry out the test in part (b)?
4. Give one factor that has not been taken into account in your analysis.

1. A glue supplier claims that Goglue is stronger than Tackfast. A company is presently using Tackfast but agrees to change to Goglue if, at the 5\% significance level,

• the standard deviation of the force required for Goglue to fail is not greater than the standard deviation of the force required for Tackfast to fail and
• the mean force required for Goglue to fail is more than 4 newtons greater than the mean force for Tackfast to fail.

A series of trials is carried out, using Goglue and Tackfast, and the glues are tested to destruction. The force, \(x\) newtons, at which each glue fails is recorded. It can be assumed that the force at which each glue fails is normally distributed.

1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the standard deviation of the force required for Goglue to fail is greater than the standard deviation of the force required for the Tackfast to fail. State your hypotheses clearly. The supplier claims that the mean force required for its Goglue to fail is more than 4 newtons greater than the mean force required for Tackfast to fail.
2. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test the supplier's claim.
3. Show that, at the \(5 \%\) level of significance, the supplier's claim will be accepted if \(\bar { X } _ { G } - \bar { X } _ { T } > 4.55\), where \(\bar { X } _ { G }\) and \(\bar { X } _ { T }\) are the mean forces required for Goglue to fail and Tackfast to fail respectively. Later, it was found that an error had been made when recording the results for Goglue. This resulted in all the forces recorded for Goglue being 0.5 newtons more than they should have been. The results for Tackfast were correct.
4. Explain whether or not this information affects the decision about which glue the supplier decides to use.

The length \(X\) mm of a spring made by a machine is normally distributed N(\(\mu, \sigma^2\)). A random sample of 20 springs is selected and their lengths measured in mm. Using this sample the unbiased estimates of \(\mu\) and \(\sigma^2\) are \(\bar{x} = 100.6\), \(s^2 = 1.5\). Stating your hypotheses clearly test, at the 10\% level of significance,

1. whether or not the variance of the lengths of springs is different from 0.9, [6]
2. whether or not the mean length of the springs is greater than 100 mm. [6]

Find the value of the constant \(a\) such that $$\text{P}(a < F_{8,10} < 3.07) = 0.94$$ [2]

A mechanic is required to change car tyres. An inspector timed a random sample of 20 tyre changes and calculated the unbiased estimate of the population variance to be 6.25 minutes². Test, at the 5\% significance level, whether or not the standard deviation of the population of times taken by the mechanic is greater than 2 minutes. State your hypotheses clearly. [6]

2. The time, \(t\) hours, that a typist can sit before incurring back pain is modelled by \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 30 typists gave unbiased estimates for \(\mu\) and \(\sigma ^ { 2 }\) as shown below. $$\hat { \mu } = 2.5 \quad s ^ { 2 } = 0.36$$

1. Find a 95\% confidence interval for \(\sigma ^ { 2 }\)
2. State with a reason whether or not the confidence interval supports the assertion that \(\sigma ^ { 2 } = 0.495\)

7. The times taken to travel to school by sixth form students are normally distributed. A head teacher records the times taken to travel to school, in minutes, of a random sample of 10 sixth form students from her school. Based on this sample, the \(95 \%\) confidence interval for the mean time taken to travel to school for sixth form students from her school is
[0pt] [28.5, 48.7] Calculate a 99\% confidence interval for the variance of the time taken to travel to school for sixth form students from her school.
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1. A diabetic patient records her blood glucose readings in \(\mathrm { mmol } / \mathrm { l }\) at random times of day over several days. Her readings are given below.

$$\begin{array} { l l l l l l l } 5.3 & 5.7 & 8.4 & 8.7 & 6.3 & 8.0 & 7.2 \end{array}$$ Assuming that the blood glucose readings are normally distributed calculate

1. an unbiased estimate for the variance \(\sigma ^ { 2 }\) of the blood glucose readings,
2. a \(90 \%\) confidence interval for the variance \(\sigma ^ { 2 }\) of blood glucose readings.
3. State whether or not the confidence interval supports the assertion that \(\sigma = 0.9\). Give a reason for your answer.

1. Two independent random samples \(X _ { 1 } , X _ { 2 } , \ldots , X _ { 7 }\) and \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 } , Y _ { 4 }\) were taken from different normal populations with a common standard deviation \(\sigma\). The following sample statistics were calculated.

$$s _ { x } = 14.67 \quad s _ { y } = 12.07$$ Find the \(99 \%\) confidence interval for \(\sigma ^ { 2 }\) based on these two samples.