An advertisement for the \"30-inch Wonder\" that appeared in the September 1983

Question

An advertisement for the "30-inch Wonder" that appeared in the September 1983 issue of the journal Packaging claimed that the 30-inch Wonder weighs cases and bags up to 110 pounds and provides accuracy to within 0.25 ounce. Suppose that a 50-ounce weight was repeatedly weighed on this scale and the weight readings recorded. The mean value was 49.5 ounces, and the standard deviation was 0.1. What can be said about the percentage of the time that the scale actually showed a weight that was within 0.25 ounce of the true value of 50 ounces? (Hint: Use Chebyshev's rule.)

We are trying to determine what percentage of the weight readings were between 49.75 ounces and ______ ounces. 49.75 ounces is ______ standard deviations above the mean value of 49.5 ounces.Chebyshev's rule tells us that at least ________% of the observations were within that many standard deviations of the mean value. So at most ______ % of the weight readings were 49.75 ounces and above.

Explanation / Answer

Ans:

We are trying to determine percentage of the weight readings were between 49.75 ounces and 49.25 ounces.

(as mean is 49.5)

49.75 is 2.5 standard deviations above the mean,

(0.25/0.1=2.5=k)

Chebyshev's rule tells us that atleast 1-(1/k^2) % of the observations were within that k standard deviations of the mean value.

here k=2.5

so, atleast 1-(1/2.5^2)=1-0.16=0.84 or atleast 84% will lie between 49.25 and 49.75

and (1-0.84)/2=0.08 or at most 8%  of the weight readings were 49.75 ounces and above.

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