Understand standardised normal variables (𝑧-values, 𝑧-scores) and use these to compare samples.

A pharmaceutical company tests the concentration of an active ingredient in two different production batches. Batch A has a concentration with mean 42.8 mg/L and standard deviation 3.2 mg/L. Batch B has a concentration with mean 38.5 mg/L and standard deviation 2.7 mg/L. A random sample from Batch A measures 48.0 mg/L, while a random sample from Batch B measures 43.1 mg/L. Explain which sample is more unusual relative to its own batch distribution. Justify your answer using standardised values.

Two independent random samples are collected from normally distributed populations. Sample P: mean \(\bar{x}_P = 72.5\), standard deviation \(s_P = 8.4\), sample size \(n_P = 35\) Sample Q: mean \(\bar{x}_Q = 68.2\), standard deviation \(s_Q = 12.6\), sample size \(n_Q = 40\) (a) Calculate the \(z\)-score for a value of \(80\) in sample P. Give your answer correct to 2 decimal places. [1 mark] (b) Calculate the \(z\)-score for a value of \(80\) in sample Q. Give your answer correct to 2 decimal places. [1 mark] (c) Using your results from parts (a) and (b), determine in which sample the value \(80\) is relatively more extreme. [1 mark] (d) Explain your reasoning in part (c) by comparing the standardised positions of the value in each sample. [1 mark]

Two students, Alex and Bailey, sit different mathematics tests. Alex's test has a mean score of 68 marks and a standard deviation of 8 marks. Bailey's test has a mean score of 72 marks and a standard deviation of 12 marks. Alex scores 84 marks and Bailey scores 90 marks. (a) Calculate the z-score for Alex's result. Express your answer correct to two decimal places. [1 mark] (b) Calculate the z-score for Bailey's result. Express your answer correct to two decimal places. [1 mark] (c) Compare the performance of Alex and Bailey relative to their respective tests, using the standardised values to support your comparison. [3 marks]

Two students sitting the same Mathematics Methods examination obtained raw scores of 72 and 68 respectively. The examination had a mean score of 70 and a standard deviation of 4 marks. Which statement correctly compares the standardised scores (z-scores) of these two students?

Two students, Aisha and Bao, each score 78 on different standardised tests. Aisha's test has a mean of 72 and a standard deviation of 4. Bao's test has a mean of 70 and a standard deviation of 5. Which statement correctly compares their standardised performance?