 # Suppose that diameters of a new species of apple have a bell-shaped distribution with a mean of 7.2cm and a standard deviation of 0.38cm. Using the empirical rule, what percentage of the apples have diameters that are less than 6.44cm? Please do not round your answer. The rule of thumb, also known as the 68-95-99.7 rule, states that under a normal distribution:

1. Approximately 68% of the data are within one standard deviation of the mean.

2. Approximately 95% of the data are within two standard deviations of the mean.

3. Approximately 99.7% of the data are within three standard deviations of the mean.

You are asking for the percentage of apples smaller than 6.44 cm in diameter. Average diameter 7.2 cm, standard deviation 0.38 cm.

Subtracting the given diameter from the average gives us:

7.2 cm - 6.44 cm = 0.76 cm

This means that 6.44 cm is two standard deviations below the average (because 0.76 cm is twice the standard deviation of 0.38 cm).

The rule of thumb is that 95% of the data are within two standard deviations of the mean. This 95% includes percentages that are both below and above average.

Since the normal distribution is symmetrical, the data is divided equally, so we divide 95% by 2 to find the percentage of the data that is below the mean minus two standard deviations. So, the percentage of apples with a diameter of less than 6.44 cm will be:

95% / 2 = 47.5%

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