13. an online retailer offers 8 different inexpensive laptop computers that are comparable in features. the…

13. an online retailer offers 8 different inexpensive laptop computers that are comparable in features. the prices quoted are $215, $130, $165, $175, $215, $160, $120, $250.\n a. determine the mean, median, and mode prices.\n b. suppose that the retailer wants the mean price of these laptops to be $170. at what price would the $250 laptop have to be set for the mean price to become $170?\n c. explain why discounting the $250 price to the amount you determined in part b only changes the mean price from part a and not the median and mode.\n14. a survey was taken of 60 amusement - park attendees who waited in line for a popular water - themed ride on july 19th. the wait times were reported in minutes as follows:\n15 30 35 35 40 40 45 45 45 50\n55 55 60 60 60 60 60 60 65 65\n65 65 70 70 70 70 70 70 70 75\n70 70 70 70 70 70 70 70 80 80\n80 80 80 80 80 80 80 90 90 90\n90 90 95 95 95 100 100 115 115\n a. find the percentile rank for a wait time of 75 minutes. interpret your results.\n b. find the percentile rank for a wait time of 40 minutes. interpret your results.\n c. based on your answers to parts a and b, which wait time would have a higher percentile rank?

13. an online retailer offers 8 different inexpensive laptop computers that are comparable in features. the prices quoted are $215, $130, $165, $175, $215, $160, $120, $250.\n a. determine the mean, median, and mode prices.\n b. suppose that the retailer wants the mean price of these laptops to be $170. at what price would the $250 laptop have to be set for the mean price to become $170?\n c. explain why discounting the $250 price to the amount you determined in part b only changes the mean price from part a and not the median and mode.\n14. a survey was taken of 60 amusement - park attendees who waited in line for a popular water - themed ride on july 19th. the wait times were reported in minutes as follows:\n15 30 35 35 40 40 45 45 45 50\n55 55 60 60 60 60 60 60 65 65\n65 65 70 70 70 70 70 70 70 75\n70 70 70 70 70 70 70 70 80 80\n80 80 80 80 80 80 80 90 90 90\n90 90 95 95 95 100 100 115 115\n a. find the percentile rank for a wait time of 75 minutes. interpret your results.\n b. find the percentile rank for a wait time of 40 minutes. interpret your results.\n c. based on your answers to parts a and b, which wait time would have a higher percentile rank?

Answer

Explanation:

Step1: Calculate the mean of laptop prices in part a

The prices are $215, $130, $165, $175, $215, $160, $120, $250. The sum is $215 + 130+165 + 175+215+160+120+250=1430$. The mean $\bar{x}=\frac{1430}{8}=178.75$.

Step2: Find the new price for the $250 laptop in part b

Let the new price of the $250 laptop be $x$. We want the new mean to be $170$. The sum of the original 7 non - $250$ prices is $1430 - 250 = 1180$. We set up the equation $\frac{1180 + x}{8}=170$. Cross - multiply: $1180+x=170\times8 = 1360$. Then $x=1360 - 1180=180$.

Step3: Explain the effect on mean, median and mode in part c

The mean is the sum of all values divided by the number of values. Changing one value changes the sum and thus the mean. The median is the middle value (or average of middle two values) when the data is ordered. Since the number of data points is 8, the median is the average of the 4th and 5th ordered values. Changing the largest value does not change the middle values. The mode is the most frequently occurring value. Changing the $250$ value does not change the most frequently occurring value(s) ($215$ in this case).

Step4: Calculate percentile rank for 75 minutes in part 14a

First, order the data. There are $n = 60$ data points. The number of values less than or equal to 75 is 39. The percentile rank $PR=\frac{39}{60}\times100 = 65$. This means that 65% of the amusement park attendees waited 75 minutes or less.

Step5: Calculate percentile rank for 40 minutes in part 14b

The number of values less than or equal to 40 is 12. The percentile rank $PR=\frac{12}{60}\times100 = 20$. This means that 20% of the amusement park attendees waited 40 minutes or less.

Answer:

a. Mean of laptop prices: $178.75 b. New price of the $250 laptop: $180 c. Changing the $250 value changes the mean because it affects the sum of values, but not the median (middle values remain the same) and mode (most - frequent values remain the same). 14a. Percentile rank for 75 minutes: 65. 65% of attendees waited 75 minutes or less. 14b. Percentile rank for 40 minutes: 20. 20% of attendees waited 40 minutes or less.