An engineering firm is testing a new solar panel design. They investigated the relationship between the energy conversion efficiency, EEE (%), and the average operating temperature, TTT (°C), for a sample of 8 test units. They also recorded a subjective durability rating, RRR, for each unit, where 1 indicates the most durable and 8 indicates the least durable.
| Unit | Efficiency EEE (%) | Durability Rating RRR |
|---|---|---|
| U1U_1U1 | 21.4 | 7 |
| U2U_2U2 | 26.8 | 2 |
| U3U_3U3 | 24.3 | 5 |
| U4U_4U4 | 27.9 | 1 |
| U5U_5U5 | 25.1 | 4 |
| U6U_6U6 | 21.1 | 8 |
| U7U_7U7 | 27.2 | 3 |
| U8U_8U8 | 22.9 | 6 |
The following summary statistics were calculated for temperature and efficiency:
STT=244.5,SEE=48.2,STE=−92.4 S_{TT} = 244.5, \quad S_{EE} = 48.2, \quad S_{TE} = -92.4 STT=244.5,SEE=48.2,STE=−92.4Calculate the product moment correlation coefficient (PMCC) between energy conversion efficiency and operating temperature.
Use your value from part (a) to test, at the 1% significance level, whether there is evidence of a negative correlation between efficiency and temperature. State your hypotheses clearly.
State a required assumption regarding the variables for the test in part (b) to be valid.
Calculate the Spearman's rank correlation coefficient between efficiency and durability rating.
Using a 5% level of significance and a two-tailed test, determine whether there is evidence of a correlation between efficiency and durability rating.
355 exam-style questions on OCR (MEI) A Level Maths 2.5 Statistical Hypothesis Testing, covering 2.5.1 Process and language of hypothesis testing, 2.5.2 When to apply 1-tail and 2-tail tests, 2.5.3 Significance level and incorrect rejection, 2.5.4 Null and alternative hypotheses (binomial), 2.5.5 Conduct a binomial hypothesis test, 2.5.6 Critical and acceptance regions (binomial), 2.5.7 Distribution of the sample mean (A-level only), 2.5.8 Hypothesis test for a single mean (A-level only), 2.5.9 Critical and acceptance regions (mean) (A-level only), 2.5.10 Correlation as closeness to a straight line (A-level only), 2.5.11 Inference using a correlation coefficient (A-level only), and 2.5 Statistical Hypothesis Testing. Each one has a worked solution and a mark scheme showing where the marks go.