International Tractor Motors (ITM) had recently undertaken an ad campaign aimed at agricultural owners in a
certain country in Asia. The ad campaign was devoted to promoting the new IT-8 large specialty tractor. Four sales associates were assigned to sell in different medium to large farms throughout the country for many days. One associate was assigned to the northern part of the country. Another was assigned to the southern part. The other two were assigned to the west and east, respectively. Because of regulations, a sales associate can sell at most one IT-8 tractor per day and only one such tractor per farm, too. A sales associate was successful if he/she sold a tractor during the day. Thus, ITM can sell at most 4 tractors (4 total sales) per day in this country.
Download the file titled Tractor Successes. It contains a scatter plot of the number of successes versus frequency. To compare the results to the Binomial Distribution, complete the following:
- Explain why this tractor sales scenario can be a binomial experiment.
- Using the Tractor Successes scatter plot, construct a frequency distribution for the number of successes.
- Compute the mean number of successes. The formula for the mean is as follows: ∑(x⋅f)∑f
The terms x represent the total number of successes (0, 1, 2, 3, 4) and f is the corresponding frequency (number of days where x successes occurred).
Explain what the numerical result means.
- From the frequency distribution, construct the corresponding relative frequency distribution.
Explain why the relative frequency distribution table is a probability distribution.
Then, use Excel for scatter plot of the probability distribution:
Select the two columns of the probability distribution. Click on INSERT, and then go to the Charts area and select Scatter. Then choose the first Scatter chart (the one without lines connecting).
- Using the frequency distribution, what is the tractor sales success average? In part 3, note that the numerator in the formula for the mean is the total number of successes. The total number of trials is the denominator of the formula for the mean multiplied by 4. What does this average mean?
- The Binomial Distribution is uniquely determined by n, the number of trials, and p, the probability of "success" on each trial. Using Excel, construct the Binomial Probability Distribution for four trials, n, and probability of success, p, as the tractor sales success average in part 5. Here is an explanation of the BINOM.DIST function in Excel: https://support.office.com/en-ie/article/BINOM-DIST-function-c5ae37b6-f39c-4be2-94c2-509a1480770c?ui=en-US&rs=en-IE&ad=I (Links to an external site.)
For example, In Excel
represents the probability of 7 successes out of 15 (n) trials. The 0.7 is the probability of success, p.
Using the above value of n=4 with probability of success, p, as the tractor sales success average in part 5, what is the probability of at least two successes?
- Using the formula for the mean of the Binomial Distribution, what is the mean number of successes in part 6 above?
- In Excel, build scatter plot for the Binomial Distribution. The instructions for creating a scatter plot are in part 4 above.
- Use the results above to compare the probability distribution of tractor sales successes and the Binomial Distribution. Compare the means in parts 4 and 6, too.
If the probability distribution of tractor sales successes and the Binomial Distribution differ, explain why that is so.
- Do you think the Binomial Distribution is a good model for the tractor sales success scenario? Why or why not?
- How can International Tractor Motors use the Binomial Distribution to approximate tractor sales in similar countries?
- In what other scenarios can International Tractor Motors use the Binomial Distribution?
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