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# Drawing Conclusions from Examples

Drawing Conclusions from Examples
• Calculus • Decision Maths • Geometry • Mechanics Maths • Probability and Statistics • Pure Maths • Statistics We will now learn how to go about drawing statistical conclusions from an experiment. The data has been collected and analysed. Perhaps it has also been visualised using a plot or a histogram, for example. The final stage, and what we have been working towards the entire time, is being able to bring all of these elements together so that a logical conclusion can be drawn from the data.

## The stages of a research experiment

In order to see how we can go about drawing conclusions from our data, we need to be keeping a few things in mind throughout the process. First, we need to think about what we expect to see from the data. Our knowledge of the subject matter and the data itself should help us decide how to go about our analysis. After this, we will be in a good position to say whether our analysis either confirms or denies our original expectations about the data.

### The hypothesis

The very first thing a researcher (that's you!) does - before the experiment and certainly before analyzing the experimental data - is come up with a hypothesis. This is a very specific prediction about what we might expect the data to show. It is important to set up the hypothesis before we analyze the data since this will determine the ways in which we look at the data.

Often, the hypothesis will be defined for you in the question, but it is still useful to keep it in mind throughout the analysis and conclusion.

Hypothesis: a specific prediction about the outcome from an experiment that is used as the starting point for research. It is usually either proven or disproven by the end of an experiment.

Say we wanted to observe the effect of sunny weather on the revenue of a lemonade stand. We might define our hypothesis to be: the revenue of the lemonade stand is higher when the sun is out.

Using common sense, people are more likely to want to buy cold drinks on sunny days, and this will probably affect the revenue. Without any more information about the situation, there is nothing else we can go by.

### The experiment and analysis

As per the example, say we collected the following data:

 Revenue, r (dollars) Frequency when sunny Frequency when not sunny $0\le x<15$ 2 52 $15\le x<30$ 3 49 $30\le x<40$ 17 29 $40\le x<50$ 29 27 $50\le x<60$ 57 17 $60\le x<70$ 62 8 $70\le x\le 90$ 30 0

Since the data is grouped, we can plot each of the sets of data using a histogram. Histogram 1, StudySmarter Originals Histogram 2, StudySmarter Originals

From the graphs, we can see that the mode class for the sunny data is the $60\le x<70$ class, whereas for the non-sunny weather the mode class is $0\le x\le 15$.

We can also find the mean as an additional measure of central tendency. Recall that to find the mean from grouped data, we need to use the midpoints of each class.

 Revenue, x (dollars) Frequency when sunny, ${f}_{1}$ Frequency when not sunny, ${f}_{2}$ Class midpoint, m m$×{f}_{1}$ m$×{f}_{2}$ $0\le x<15$ 2 52 7.5 15 390 $15\le x<30$ 3 49 22.5 67.5 1,102.5 $30\le x<40$ 17 29 35 595 1,015 $40\le x<50$ 29 27 45 1,305 1,215 $50\le x<60$ 57 17 55 3,135 935 $60\le x<70$ 62 8 65 4,030 520 $70\le x\le 90$ 30 0 80 2,400 0

Now we can find the mean for the 'sunny days' revenue:

$\overline{){r}_{1}}=\frac{\sum _{}\mathrm{m}×{f}_{1}}{\sum _{}{f}_{1}}=\frac{15+67.5+595+1305+3135+4030+2400}{5+17+29+57+62+30}=\frac{11547.7}{200}=57.74$ to 2 d.p.

And now find the mean for the 'non-sunny days' revenue:

$\overline{){r}_{1}}=\frac{\sum _{}\mathrm{m}×{f}_{2}}{\sum _{}{f}_{2}}=\frac{390+1102.5+1015+1215+935+520+0}{52+43+35+27+17+8+0}=\frac{5252.5}{182}=28.86$ to 2 d.p.

### The conclusion

So, we have collected and analyzed the data. All that is left to do is to compare our statistics with the hypothesis we made beforehand.

Recall: the revenue of the lemonade stand is higher when the sun is out.

Now we compare our original hypothesis with the statistics we have found. We can see that the average amount of revenue on a sunny day is $59.95, while the average revenue on a day that is not sunny is$18.24. Since the sunny-day revenue is so much higher than otherwise, we can conclude that the data we have collected supports our hypothesis and that, according to the data, the revenue of the lemonade stand is higher when the sun is out.

This process - drawing a conclusion about a population based on results collected from a sample - is called statistical inference.

But beware! It is important to be careful of the language we use in our hypothesis. Take for example the following statement: "sunny weather makes people more likely to buy lemonade". While this may be true, we don't know enough from the data itself to confirm that it was the weather specifically that inspired more people to buy lemonade. Instead, it could have been the case that the weather increased the number of potential customers in the vicinity of the lemonade stand.

We can also use other statistics to add to our conclusion. We can see that the range of the 'sunny day' revenues is 90 whereas the range of the 'not sunny days' is 70. Therefore we can also add to our conclusion that while there is on average a greater amount of revenue on sunny days, there is also a greater range in the revenue.

Your conclusion must always refer specifically to what the data shows us!

## Drawing Conclusions from Examples - Key takeaways

• The hypothesis is a specific prediction about the outcome of an experiment that is used as the starting point for research. It is usually either proven or disproven by the end of an experiment
• To draw conclusions, we first must collect the relevant data and perform statistical analyses such as creating visualizations of the data (e.g. a histogram) and finding relevant statistics (e.g. the mean)
• In the conclusion, compare the statistical analysis with your original hypothesis
• The process of drawing a conclusion about a population based on results collected from a sample is called statistical inference
• Be careful in your conclusion to only refer to what the statistics specifically tell us about the data

If the average time spent doing math homework is higher than the average time doing biology homework, we draw the following conclusion: 'on average, students spend more time doing math homework than biology homework'.

In the conclusion of an experiment, you should compare your original hypothesis with statistics you have derived from the data.

Drawing a conclusion means using statistics to make a statement about the thing you have measured in the experiment. The process of drawing a conclusion about a population based on results collected from a sample is called statistical inference.

## Final Drawing Conclusions from Examples Quiz

Question

Misleading graphs are the graphs that provide incorrect conclusions by misinterpretation of data.

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Question

Are misleading graphs always created unintentionally?

No, they are also created intentionally to mislead audience.

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Question

Identify the way to create misleading graphs.

Data usage

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Question

What is a hypothesis?

A hypothesis is a specific prediction about the outcome from an experiment that is used as the starting point for research. It is usually either proven or disproven by the end of an experiment.

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Question

Which of the following are true?

A hypothesis should be specific

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Question

What are the three stages of a research experiment?

1. Hypothesis

2. Experiment and analysis

3. Conclusion

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Question

Say you were planning on collecting data on the heights of sunflowers grown under two different conditions: inside and outside. What sensible hypothesis might we draw from this?

A possible hypothesis: the sunflowers grown outside will be taller than the sunflowers grown inside.

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Question

What is statistical inference?

It is the process of drawing a conclusion about a population based on results collected from a sample.

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Question

Which of the following are true?

Statistical inference is where you make conclusions about a sample based off information about the population.

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What is a conclusion?

The conclusion is the final stage in a research experiment where we decide whether our hypothesis is valid or not using statistical inference.

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What is characteristic of a good conclusion?

The conclusion must always refer to what the data shows and we must be careful not to make statements that aren't supported by the statistics collected.

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Question

Say you were planning on collecting data on the amount of time spent completing homework assignments for two different subjects: math and chemistry. What sensible hypothesis might we draw from this?

There are a number of possible right answers for this question. The following is a possible hypothesis you may choose: the time spent completing homework is longer for chemistry than it is for math.

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Question

Say you were planning on collecting data on the lifespan of batteries in mobile phones of two brands, Pear and Raspberry. What sensible hypothesis might we draw from this?

Given that we don't have any extra information or anecdotal information about the mobile phone brands, the following might be a good hypothesis: the lifespan of batteries in Pear and Raspberry phones is on average the same.

Our hypothesis would be disproven if one brand's phones lasted significantly longer than the other.

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Question

Which of the following are true?

The conclusion refers to what the data shows.

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Question

Suppose we collected data on the total number of pets owned by people that own at least one cat, x, and the number of pets owned by people that own at least one dog, y

From the data, we have the following statistics:

Meanx=3.57

Meany=1.40

Which of the following are true.

From the data presented, if someone owns one cat, they are more likely to own more pets.

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Question

Suppose we collected data on the number of clothes sold daily in a shop during summer months, x, and the number of clothes sold in during the winter months, y.

From the data, we have the following statistics:

Meanx=180.6

Variancex=15.2

Meany=187.2

Variancey=34.9

Which of the following are true.

During winter months, there is a greater dispersion of the number of clothes sold daily than in the summer months.

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Question

'We can always come to a conclusion using statistics from our data.'

Explain why this may not be true.

It may be the case that the statistics we come up with are too similar to come to a definitive conclusion.

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Question

Suppose we collected data on the growth of plants that were played classical music, x, and the growth of plants that were played rock music, y.

From the data, we have the following statistics:

Meanx=12.29cm

Variancex=1.34cm

Meany=12.78cm

Variancey=1.87cm

What can we say about the growth of both groups of plants in relation to these statistics?

Although the growth of the classical music group was less on average than the growth of the rock music group, the difference is very little, so there isn't much evidence to suggest one group grows faster than the other.

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Question

Misleading graphs are also called as?

Organized graphs

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Question

If proper scaling in a graph is used then are they considered misleading?

Yes

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Question

3D graph

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Question

Can a misleading graph be used to present correct information?

No

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Question

State any one step to rectify misleading graphs.

Change the scaling of the graph if it does not start from 0.

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Question

"If the intervals on both axes are not even, construct a new graph with even intervals."

No

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Question

In pictographs, proper key and symbol size is most important.

True

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Question

Give one reason why a misleading graph is used.

To pursue the audience.

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Question

Is it possible to rectify any misleading graphs?

Yes

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