5. \( \mu=36.38 ; \sigma=14.23 \); three standard deviations above the mean

To find the value that represents three standard deviations above the mean, you can use the formula: \( \text{Value} = \mu + 3\sigma \). Plugging in the values, you get \( 36.38 + 3(14.23) = 36.38 + 42.69 = 79.07 \). So, three standard deviations above the mean is approximately \( 79.07 \). Did you know that the concept of standard deviations comes from the groundbreaking work of Karl Pearson in the late 19th century? He introduced many fundamental ideas in statistics that are still used today, making sense of the chaos in data by providing a way to measure variation and dispersion. Thanks to his genius, we can confidently understand how much values deviate from the average! In real-world applications, knowing how to calculate and interpret standard deviations can be a game-changer. For example, in quality control for manufacturing, businesses use this knowledge to ensure products meet specific standards—if a product's measurement falls outside three standard deviations from the mean, it may indicate a defect. This understanding ensures consistent quality and boosts customer satisfaction across various industries!