State suitable hypotheses for a two-tailed test.

Find the critical region for testing $\overline{b}$ at the 5 % significance level.

Find the probability of making a Type II error.

Another model of smartphone whose battery life may be assumed to be normally distributed with mean μ hours and standard deviation 1.2 hours is tested. A researcher measures the battery life of six of these smartphones and calculates a confidence interval of [10.2, 11.4] for μ.

Calculate the confidence level of this interval.

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

Note: In question 3, accept answers that round correctly to 2 significant figures.

${\text{H}}_0\text{:}\mu =9.5\text{;}{\text{H}}_1\text{:}\mu \ne 9.5$     A1

[1 mark]

Note: In question 3, accept answers that round correctly to 2 significant figures.

the critical values are $9.5\pm 1.95996\dots \times \frac{0.4}{\sqrt{20}}$     (M1)(A1)

i.e. 9.3247…, 9.6753…

the critical region is $\overline{b}$ < 9.32, $\overline{b}$ > 9.68     A1A1

Note: Award A1 for correct inequalities, A1 for correct values.

Note: Award M0 if t-distribution used, note that t(19)_97.5 = 2.093 …

[4 marks]

Note: In question 3, accept answers that round correctly to 2 significant figures.

$\overline{B}\sim \text{N}\left(9.8,{\left(\frac{0.4}{\sqrt{20}}\right)}^2\right)$     (A1)

     (M1)

=0.0816     A1

Note: FT the critical values from (b). Note that critical values of 9.32 and 9.68 give 0.0899.

[3 marks]

Note: In question 3, accept answers that round correctly to 2 significant figures.

METHOD 1

$X\sim \text{N}\left(\text{10}\text{.8,}\frac{{1.2}^2}{6}\right)$     (M1)(A1)

P(10.2 < X < 11.4) = 0.7793…     (A1)

confidence level is 77.9%    A1

Note: Accept 78%.

METHOD 2

$11.4-10.2=2z\times \frac{1.2}{\sqrt{6}}$      (M1)

$z=1.224\dots$     (A1)

P(−1.224… < Z < 1.224…) = 0.7793…      (A1)

confidence level is 77.9%      A1

Note: Accept 78%.

[4 marks]