Question

mcdonalds has 13 different hamburgers on the menu. the prices are displayed in the dot - plot and summary statistics are provided.
a mcdonalds manager in michigan, where there is a sales tax of 6%, adjusts all these prices to include this sales tax.
a. what is the price that includes sales tax for the big - mac priced at $4.49? what mathematical operation was used for this calculation?
b. what is the shape of the distribution of mcdonalds hamburger prices that include sales tax?
c. what are the mean and standard deviation of the distribution of mcdonalds hamburger prices that include sales tax?
d. what are the median and iqr of the distribution of mcdonalds hamburger prices that include sales tax?

Explanation:

Step1: Calculate price with tax for Big - Mac

The formula to find the price including tax is $P = P_0(1 + r)$, where $P_0$ is the original price and $r$ is the tax rate. Given $P_0 = 4.49$ and $r=0.06$. So $P = 4.49\times(1 + 0.06)=4.49\times1.06 = 4.7594\approx4.76$. The operation is multiplication.

Step2: Analyze shape of distribution

When each data - point in a distribution is multiplied by a positive constant ($1 + 0.06=1.06$ in this case), the shape of the distribution remains the same.

Step3: Calculate new mean and standard deviation

If each data - point $x_i$ in a distribution is multiplied by a constant $k$, the new mean $\mu_{new}=k\mu_{old}$ and the new standard deviation $\sigma_{new}=k\sigma_{old}$. Here $k = 1.06$, $\mu_{old}=2.95$, so $\mu_{new}=1.06\times2.95 = 3.127\approx3.13$ (there was an error in the hand - written answer for the mean). $\sigma_{old}=1.53$, so $\sigma_{new}=1.06\times1.53 = 1.6218\approx1.62$.

Step4: Calculate new median and IQR

If each data - point is multiplied by a constant $k$, the new median $M_{new}=kM_{old}$ and the new IQR $IQR_{new}=kIQR_{old}$. Given $M_{old}=3.69$, so $M_{new}=1.06\times3.69 = 3.9114$. $IQR_{old}=Q_3 - Q_1=4.49 - 1.34 = 3.15$, so $IQR_{new}=1.06\times3.15 = 3.339$.

Answer:

a. The price that includes sales tax for the Big - Mac is $\$4.76$. The mathematical operation used is multiplication.
b. The shape of the distribution stays the same.
c. Mean: $\$3.13$, Standard deviation: $\$1.62$
d. Median: $\$3.9114$, IQR: $\$3.339$