The National Sporting Goods Association (NSGA) conducted a survey of the ages of individuals that purchased skateboarding footwear. The ages of this survey are summarized in the following relative frequency distribution. Suppose the survey was based on a sample of 200 individuals. Age of User Percent Under 14 years old 35 14 to 17 years old 41 18 to 24 years old 15 25 to 34 years old 4 35 to 44 years old 4 45 to 64 years old 1 Calculate the mean, variance and standard deviation of the age of individuals that purchased skateboarding shoes. Use 10 as the midpoint of the first class. (Round your answers to 2 decimal places.)

[tex]\left[\begin{array}{cc}Age\ of\ user&Percent\\Under\ 14\ years\ old&35\\14\ to\ 17\ years\ old&41\\18\ to\ 24\ years\ old&15\\25\ to\ 34\ years\ old&4\\35\ to\ 44\ years\ old&4\\45\ to\ 64\ years\ old&1\end{array}\right][/tex]

The mean is the sum of the midpoints times the percentage.

μ = ∑ (mx)

The variance is the sum of the squared differences between the midpoints and the mean, all divided by the total number of people minus one.

σ² = ∑ (m − μ)² / (n − 1)

And the standard deviation is the square root of the variance.

First, we calculate the mean.

[tex]\left[\begin{array}{cccc}Age\ of\ user&Percent&m&mx\\Under\ 14\ years\ old&35&10&3.5\\14\ to\ 17\ years\ old&41&15.5&6.355\\18\ to\ 24\ years\ old&15&21&3.15\\25\ to\ 34\ years\ old&4&29.5&1.18\\35\ to\ 44\ years\ old&4&39.5&1.58\\45\ to\ 64\ years\ old&1&54.5&0.545\\&&\mu&16.31\end{array}\right][/tex]

Now, we find the variance.

[tex]\left[\begin{array}{cccc}Age\ of\ user&Percent&m&(m-\mu)^{2}\\Under\ 14\ years\ old&35&10&39.82\\14\ to\ 17\ years\ old&41&15.5&0.66\\18\ to\ 24\ years\ old&15&21&22.00\\25\ to\ 34\ years\ old&4&29.5&173.98\\35\ to\ 44\ years\ old&4&39.5&537.78\\45\ to\ 64\ years\ old&1&54.5&1458.48\\&&\sigma^{2}&11.22\end{array}\right][/tex]

The standard deviation is therefore σ = 3.35.