The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 44 ounces and a standard deviation of 10 ounces. Use the Empirical Rule. Suggestion: sketch the distribution in order to answer these questions. a) 95% of the widget weights lie between 34 and 54 b) What percentage of the widget weights lie between 34 and 64 ounces

a) No 95% of values will fall between (24; 64); 68,27% will fall between (34; 54)

b) 71,83 % will fall between 34 and 64 ounces

Step-by-step explanation:

Empirical rule establishes, for a normal distribution with mean μ and σ as standard deviation:

In interval μ ± σ or (μ + σ; μ - σ) we should find 68.27 % of all values of the population, and by simmetry 68.27/2 = 34,14 % should be over the mean and the other half would be values below the mean

Therefore in our case

μ + σ = 44 + 10 = 54

And

μ - σ = 44 - 10 = 34

a) Then 68,34 % of values will fall in this interval

We know now that value 34 is 1 * σ below the mean, and is at the limit of 34,14 %

b) μ + 2*σ = 44 * 2*10 = 44 + 20 = 64

64 is the upper limit for the interval μ + 2*σ and we know that 95.45 % of all values will fall between (μ - 2*σ; μ + 2*σ) and by simmetry just one side of this interval (the right side) will have 95.45/2 = 47; 73 %

Then in interval going from (34; 64) we shoud find 47.73 + 34,14

71,83 % of all values will fall between 34 and 64