An acoustic engineer suspects that as the Signal-to-Noise Ratio (SSS, measured in dB) of a digital transducer increases, the Subjective Clarity Rating (RRR, measured on a scale of 1 to 10) also increases. The engineer tests five prototype transducers, α,β,γ,δ \alpha, \beta, \gamma, \delta\,α,β,γ,δ and ϵ\epsilonϵ, and records the following metrics:
TransducerαβγδϵSignal-to-Noise Ratio S (dB)18.225.415.122.812.5Subjective Clarity Rating R4.37.95.66.83.9 \begin{array}{|l|c|c|c|c|c|} \hline \text{Transducer} & \alpha & \beta & \gamma & \delta & \epsilon \\ \hline \text{Signal-to-Noise Ratio } S \text{ (dB)} & 18.2 & 25.4 & 15.1 & 22.8 & 12.5 \\ \hline \text{Subjective Clarity Rating } R & 4.3 & 7.9 & 5.6 & 6.8 & 3.9 \\ \hline \end{array} TransducerSignal-to-Noise Ratio S (dB)Subjective Clarity Rating Rα18.24.3β25.47.9γ15.15.6δ22.86.8ϵ12.53.9Calculate Spearman's rank correlation coefficient for this sample.
Stating your hypotheses clearly, test at the 5% level of significance whether these data provide evidence to support the engineer's suspicion.
316 exam-style questions on OCR A Level Maths 2.5 Statistical Hypothesis Testing, covering 2.5.1 The language of hypothesis testing, 2.5.2 Hypothesis test for a binomial proportion, 2.5.3 Inference and significance level, 2.5.4 Sample mean as a random variable (A-level only), 2.5.5 Hypothesis test for the mean of a normal distribution (A-level only), 2.5.6 Pearson's product-moment correlation coefficient, and 2.5.7 Hypothesis test using Pearson's coefficient (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.