5 The mass of peaches sold per day in a supermarket is normally distributed with mean 65.8kg and standard deviation 9.6kg. (a) Find the probability that the mass of peaches sold on any given day is between 56kg and 75 kg. [3]

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We have a normal distribution problem about peach sales in a supermarket. The mass of peaches sold per day follows a normal distribution with mean 65.8 kilograms and standard deviation 9.6 kilograms. We need to find the probability that the mass sold on any given day is between 56 and 75 kilograms. This bell-shaped curve represents our normal distribution, with the mean at 65.8 kg marked by the red dashed line. To solve this problem, we need to standardize our values using z-scores. The z-score formula is z equals x minus mu divided by sigma. For x equals 56 kilograms, we calculate z1 equals 56 minus 65.8 divided by 9.6, which equals negative 9.8 divided by 9.6, giving us z1 equals negative 1.02. For x equals 75 kilograms, we calculate z2 equals 75 minus 65.8 divided by 9.6, which equals 9.2 divided by 9.6, giving us z2 equals 0.96. This transforms our original normal distribution into the standard normal distribution. Now we need to find the probability that Z is between negative 1.02 and 0.96. This equals P of Z less than 0.96 minus P of Z less than negative 1.02. Using the standard normal table, we find that P of Z less than 0.96 equals 0.8315, and P of Z less than negative 1.02 equals 0.1539. The shaded green area represents the probability we're looking for, which is the area between these two z-values. Now we complete our calculation. 0.8315 minus 0.1539 equals 0.6776. This can be expressed as a decimal 0.6776 or as a percentage 67.76%. The interpretation is that there is approximately a 67.76% chance that peach sales will be between 56 kilograms and 75 kilograms on any given day. Both the original distribution and the standard normal distribution show the same shaded area representing this probability of 0.6776. Let's summarize our complete solution. Step 1: We identified the parameters with mean 65.8 kilograms and standard deviation 9.6 kilograms. Step 2: We calculated the z-scores, getting z1 equals negative 1.02 for 56 kilograms and z2 equals 0.96 for 75 kilograms. Step 3: We found the probabilities from the standard normal table, with P of Z less than 0.96 equals 0.8315 and P of Z less than negative 1.02 equals 0.1539. Step 4: We calculated the final answer as 0.8315 minus 0.1539 equals 0.6776 or 67.76 percent. Therefore, the probability that peach sales are between 56 and 75 kilograms is approximately 67.76 percent.