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Answer:
68% of the diameters are between 7.06 cm and 7.78 cm
Step-by-step explanation:
Mean diameter = μ = 7.42
Standard Deviation = σ = 0.36
We have to find what percentage of diameters will be between 7.06 cm and 7.78 cm. According to the empirical rule, for a bell-shaped data:
- 68% of the values are within 1 standard deviation of the mean. i.e. between μ - 1σ and μ + 1σ
- 95% of the values are within 2 standard deviations of the mean. i.e. between μ - 2σ and μ + 2σ
- 99.7% of the values are within 3 standard deviation of the mean. i.e. between μ - 3σ and μ + 3σ
So, we first need to find how many standard deviations away are the given two data points. This can be done by converting them to z-score. A z score tells us that how far is a data value from the mean. The formula to calculate the z-score is:
x = 7.06 converted to z score will be:
x = 7.78 converted to z score will be:
This means the two given values are 1 standard deviation away from the mean and we have to find what percentage of values are within 1 standard deviation of the mean.
From the first listed point of empirical formula, we can say that 68% of the data values lie within 1 standard deviation of the mean. Therefore, 68% of the diameters are between 7.06 cm and 7.78 cm