State suitable hypotheses for a two-tailed test.

Find the critical region for testing \({\bar b}\) 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.

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 \(9.5 \pm 1.95996 \ldots  \times \frac{{0.4}}{{\sqrt {20} }}\)     (M1)(A1)

i.e. 9.3247…, 9.6753…

the critical region is \({\bar b}\) < 9.32, \({\bar b}\) > 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.

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

\({\text{P}}\left( {9.3247 \ldots  < \bar B < 9.6753 \ldots } \right)\)     (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)

\(z = 1.224 \ldots \)     (A1)

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

confidence level is 77.9%      A1

Note: Accept 78%.

[4 marks]