Investigation into a driving test.

By looking through the data there seems to be an awful lot of entries. I will now draw a stem and leaf diagram for the number of minor mistakes against frequency. The stem and leaf diagram is shown overleaf.  

I am now going to test some hypotheses using the stem and leaf diagram.

Hypothesis 1:  Men perform better than females in the driving test.

For this hypothesis I would like to narrow down the number of entries. The steps I will take to do this are as follows.

• Sort the original database by gender (it’s already been sorted by instructor)
• Insert a column at the start i.e. before the gender column numbering each group, e.g. instructor A, males, 1-29 etc.

From sorting out this data I can now make out the following table.

We have already said that the sample is too large to deal with. There are 240 pupils in the sample, so we will now stratify the sample. We will just tale 80 pupils i.e. a third of each of the entries above. Doing it his way we will ensure we have proportional representation. Therefore we have the following numbers.

We picked the numbers in each group by random sampling. We then copied and pasted our eighty random pupils into a new spreadsheet. We sorted this new spreadsheet by gender and the number of minor mistakes. The resulting spreadsheet is overleaf.

Using the table I will now make out a grouped frequency table for both genders.

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I will now make out a cumulative frequency table for both genders.

For these cumulative frequency tables I will draw two separate cumulative curves, i.e. 1 for Females and 1 for Males, as shown over leaf.

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I will no work out the median, lower quartile and upper quartile for each group of each gender. For the females the median will be the number of mistakes of the 21st pupil.

The median for the males will be the number of minor mistakes for the 20th candidate.

The lower quartile for the females will be the number of minor mistakes of the ¼ (41)st  = 10.25th.

The upper quartile for the females will be the number of minor mistakes of the ¾ (41)st = 30.74th.

For the males, the lower quartile will be the number of minor mistakes of the

¼ (39)th = 9.75th.

The upper quartile for the males will be the number of minor mistakes of the

¾ (39)th = 29.25th.

For Females: Median = 18

                                                                 L.Q. = 11.5

                                                                 U.Q. = 26.5

For Males: Median = 11.5

                                                               L.Q. = 6

                                                               U.Q. = 21

Now that we know the median and quartiles for each group, we can draw their two box plots.

                                                11.5          18        26.5

Females:        

                           0          10            20           30         40

        

         6         11.5          21

Males:        

 0          10            20           30         40

On examining the two box plots I can see clearly that the box for the males is more to the left than the females. That means on average then the males made fewer mistakes than females in their driving test, thus proving that my hypothesis is correct.

Hypothesis 2: The more lessons a pupil takes the less number of mistakes they make

To do this I will sort out my last spreadsheet by instructor, as shown below.

I will now draw a scatter graph of number of 1-hour lessons against number of minor mistakes for each instructor.

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By studying the four graphs, I see that for the graph for Instructor A and B, the results aren’t as conclusive and there is really no correlation and the graphs for instructor C and D were more definite. The reason that the graphs for instructor A and B weren’t as conclusive was probably due to the sample taken to plot the graphs or it may have been down to the instructor. However by looking my graph for C and D I can safely say that the more lessons a pupil takes the less number of mistakes they make.

Hypothesis 3: - There are more mistakes towards the end of the week.

For this, I will sort out the original spreadsheet by day, and then I will calculate the total number of minor mistakes per lesson for each pupil. After working this out, here are the results I found:

Total number of minor mistake per lesson for Monday = 43

Total number of minor mistake per lesson for Tuesday = 28

Total number of minor mistake per lesson for Wednesday = 37

Total number of minor mistake per lesson for Thursday = 60

Total number of minor mistake per lesson for Friday = 47

I will now calculate the average number of minor mistakes per lesson; to do this I will divide the total number of minor mistakes by the number of Mondays, Tuesdays etc. Here are the results I found:

Average number of minor mistake per lesson for Monday = 0.87

Average number of minor mistake per lesson for Tuesday = 0.85

Average number of minor mistake per lesson for Wednesday = 0.99

Average number of minor mistake per lesson for Thursday = 1.09

Average number of minor mistake per lesson for Friday = 0.75

I will now draw up a bar graph showing the average number of minor mistakes against the day of the week, as shown over leaf. Bt looking at my results you can see that as the week progresses the number of minor mistakes increases apart from until it gets to Friday, thus proving that my hypothesis is incorrect.

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Hypothesis 4: - Some instructors are better than others.

For this hypothesis I will refer back to the original data sheet and a work out the total number of minor mistakes per lesson for each instructor. From doing this I found that:

Total number of minor mistakes for Instructor A = 57

Total number of minor mistakes for Instructor B = 80

Total number of minor mistakes for Instructor C = 35

Total number of minor mistakes for Instructor D = 42

I will now calculate the average number of minor mistakes for the instructors. I will do this by dividing the total number of minor mistakes by the amount of instructor A, B, C and D. Here is what I found:

Average number of minor mistakes for Instructor A = 0.97

Average number of minor mistakes for Instructor B = 0.81

Average number of minor mistakes for Instructor C = 0.90

Average number of minor mistakes for Instructor D = 1.08

I will now draw bar chart show the average number of minor mistakes against instructor, as shown overleaf. From looking at my results and bar chart I can see that there are less mistakes made when the pupils are with instructor C and there are more mistakes made with instructor B. As there is a great difference in all of the instructors I can safely say that some instructors are better that others therefore proving that my hypothesis is correct.