In the past, the mean time taken by Freda for a particular daily journey was $$\(39.2\)$$ minutes. Following the introduction of a one-way system, Freda wishes to test whether the mean time for the journey has decreased. She notes the times, $$\(t\)$$ minutes, for 40 randomly chosen journeys and summarises the results as follows. $$\[ n=40 \quad \Sigma t=1504 \quad \Sigma t^{2}=57760 \]$$ Test, at the $$\(5 \%\)$$ significance level, whether the population mean time has decreased. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_m20_qp_62 Year：2020 Question No：3(b)
Answer:
\(\mathrm{H}_{0}:\) Pop mean \((\) or \(\mu)=39.2\)
\(\mathrm{H}_{1}\) : Pop mean \((\) or \(\mu)<39.2\)
\(\begin{array}{l}
\frac{37.6 \text { '-39.2 }}{\frac{\sqrt{131.0154 '}}{\sqrt{40}}}
=-1.817
' 1.817 '>1.645 \quad \mathrm{OE}
\end{array}\)
There is evidence that mean time has decreased
\(\mathrm{H}_{1}\) : Pop mean \((\) or \(\mu)<39.2\)
\(\begin{array}{l}
\frac{37.6 \text { '-39.2 }}{\frac{\sqrt{131.0154 '}}{\sqrt{40}}}
=-1.817
' 1.817 '>1.645 \quad \mathrm{OE}
\end{array}\)
There is evidence that mean time has decreased
Knowledge points:
6.5.3 formulate hypotheses and carry out a hypothesis test concerning the population mean in cases where the population is normally distributed with known variance or where a large sample is used
Solution:
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