A manufacturer of high-precision capacitors specifies that the mean capacitance of a specific model is 14.50 μF14.50\, \mu \text{F}14.50μF. A quality control engineer believes that a recent manufacturing fault has resulted in a lower mean capacitance than specified. A random sample of 81 capacitors is tested, yielding a sample mean of 14.38 μF14.38\, \mu \text{F}14.38μF and a standard deviation of 0.45 μF0.45\, \mu \text{F}0.45μF. Test the engineer's belief at the 1% level of significance. State your hypotheses clearly.
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.