Example A tattoo parlor that does piercings offers them to customers as Category I, II, or III, and cost $30, $50, and $70 for stainless steel jewelry, respectively, and $52, $80, and $110 for gold. The parlor sold five Category I, two Category II, and three Category III in stainless, and one each of categories I, II, and III in gold in a month. It costs the parlor $1,000 per month to offer piercings. Complete parts 1. through 4. below. 1. Find the mean for the sales. The mean is the sum of the sales divided by the number of sales. Use the formula below to find the mean of the sales. Mean = sum of values Number of values Find the sum of the sales. Remember that there are five $30 sales, two $50, sales, three $70 sales, one $52 sale, one $80 sale, and one $110 sale. Find the number of values. $110+$80+$70+$70+$70+...+$30+$30+$30+$30+$30 = 702 Find the number of values Number of values = 13 Use the information found and the formula provided to find the mean of the sales. Round to the nearest cent. Mean = sum of values number of values = $702 13 = $54.00 Find the median for the sales. Arrange the sales in order from smallest to largest or largest tosmallest, then count the number of sales. If the number of sales is odd, identify the sale in the middle. If the number of sales is even, find the mean of the middle two sales. The median is the middle sale or the mean of the middle two sales. Arrange the sales in order from largest to smallest. Enter the missing sales. $110 $80 $70 $70 $70 $52 $50 $50 $30 $30 $30 $30 $30 Since there are an odd number of sales, the middle sale is the median. Use this information and the ordered list to find the median. Median =$50 Find the mode(s) for the sales. For each sale, count the number of times the sale occurs. Identify the sale or sales that occur most frequently. The mode(s) is/are the most frequent sale(s). Find any sale(s) that occur more than once, then find the most frequent of those sale(s). Select the correct choice below and fill in any answer boxes within your choice. Mode = $30 2. Given the total sales value for the parlor's first month, how long will it take for the business to breakeven, assuming a 10 percent increase in sales value each month? Increase the monthly earnings by 10% until the total is over $1000. The number of times that the earning must be increased is the number of months it takes to reach that level. Find the value of the earnings after each increase of 10% $ 702.00 * 110% $702.20 $ 772.20 * 110% = 849.42 $ 849.42 * 110%= $934.36 $ 934.36 *110% = 1027.80 The number of increases it took to reach $1000 is the number of months that must pass for this level of earning to be reached. Determine the number of months that must pass. Number of months = 4 Is the increase more likely to come from increased number of sales or a higher average sales value? With a current mean of only 3 sales per week, any improvement is more likely to come from increased number of sales. Note that each additional sale adds an average of $54.00 to the parlor's revenue. 3. The parlor's second month results show that it made six sales at $303, one at $50, one at $52, three at $70, three at $80, and two at $110. Calculate the standard deviation for this data set. To find the standard deviation first find the mean of the new sales, then find the deviation in the value of each sale from the mean by subtracting the mean from the price of the sale. Next, square and sum all of the deviations. Find the variance by dividing the sum of the squared deviations by n minus 1n1, where n is the number of sales. Lastly, take the square root of the variance to find the standard deviation. Start by finding the mean of the sales from the second month. Find the sum of the sales. Sum of values = $110 + $110 + $80 + $80 + $80 +...+#30 +$30 + $30 + $30 +$30 = 952 Find the number of values. Number of values = 16 Use the information found and the formula provided to find the mean of the sales. Round to the nearest cent. Mean = = sum of values Number of values $952 = $59.50 16 Find the deviation in the value of each sale from the mean by subtracting the mean from the price of the sale, then square all of the deviations. Rather than repeat this process for each individual sale, multiply the square of the deviation for each type of sale by the number of sales made of that type, then add all the totals together to find the sum of squared deviations. Values Deviation $30 $50 $52 $7070 $80 $110 - 29.50 - 9.50 7.50 -10.50 -20.50 -50.50 Square of Deviation Total l 6 * 870.25 = 5221.50 1* 90.251= 90.25 1*56.25 = 56.25 3*110.25 = 330.75 3*420.25 = 1260.75 2*2550.25 = 5100.50 Sum of squared deviations= 12.060.00 # of occurrences * Square of Deviation 870.25 90.25 56.25 110.25 420.25 2550.25 The next step is to find the variance by dividing the sum of squared deviations by n minus 1n1, where n is the number of sales. Round to two decimal places. Variance = sum of squared deviations =Sum equals= equals= Lastly, find the standard deviation by taking the square root of the variance. Round to two decimal places. Standard Deviation = = = Root Variance Root 804.00 28.35 Does your answer for the standard deviation indicate that this is a normal distribution? If not, what are the implications? A normal distribution would have roughly 68% of the values within one standard deviation. Less than68% of the values are within one standard deviation from the mean. This means that future sales may vary greatly. 4. Month 3 was a breakthrough, as the parlor made seven sales at $30, three at $50, one at $52, five at $70, two at $80, and three at $110, The parlor also saw a marked increase in tattoo business due to direct referrals from the piercing artist, and the parlor decided to pay the artist a 15% referral fee. If the parlor had new tattoo business of $1200 from the referrals, what were the total earnings of the artist for the month? To find the total earnings for this month, add all the sales from the piercings and then add 15% of the $1200 tattoo sales for the referral fee. Find the total earnings for this month. Earnings - Sum + Referral Fee of sale Question is complete. = ($110 + $110 + $110+...+$30 + $30 + $30) + (0.15*$1200) = $1252 + 180 = $1432