Problem and Plan We need to think about how accurate our data is. The accuracy of our analysis and conclusion depends on the accuracy of the data.
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Exercise: Problem I wonder if Year 13’s tend to send more text messages per day than Year 11’s at McAuley High School? Plan We will ask 30 Year 11 students and 30 Year 13 students the following question: How many texts do you send each day? 1)
Do you think everyone who doesn’t have a phone would feel confident telling a stranger that? Explain.
2)
Who do teenagers send text messages to?
3)
Who do primary school children send text messages to?
4)
Do you think primary school and secondary school students would send about the same number of text messages per day? Why/why not? Explain.
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5)
Could we use the results of a survey of teenagers to estimate how many texts are sent per day by people in their 20’s? Why/why not? Explain. (Think about what text messages they might be sending and to whom …)
6)
This is a biased question. “Why is it cool to send text messages?” How could we change the wording of this question to get all reasons why students send text messages, and not just the “cool” reasons?
7)
Think about whether 18 year olds will always give truthful answers when asked how much alcohol they have had to drink in the last week. What answer might they give and why?
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Data and Analysis
With skewed data, the mean is always pulled in the direction of the long tail.
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Exercise: Predict what the shapes of the following situations may be: 1)
Ages of all the people in our school
Dot Plot
llection 1
0 2)
0 3)
20
40
60 skips
80
100
120
Time to get to school (students)
20
40
60 skips
80
100
120
Would you predict that the house prices in Auckland are likely to be approximately Normally distributed (lovely bell-shaped curve), or skewed (long tail due to multiple outliers)? Why?
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14
4)
Draw a picture of what shape you expect the distribution to take, and predict where the mean and median would be.
5)
So which measure of center would best predict the price most Aucklanders could expect their house to be worth, the mean or the median? Why?
6)
Now we want to form a more general rule: a)
When are the mean and median different? Why?
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7)
b)
If the mean and median are the same, which represents the center best? The mean or the median? Why?
c)
If the mean and median are different, which represents the center best? The mean or the median? Why?
What about measures of spread. When should you use the range? When should you use the IQR? Explain.
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Writing Comparative Statements Here are the features you can add to your analysis (shape, center, spread). 4. Comparing the Middle 50% 5. Unusual features
4. Comparing the Middle 50% Look at the boxes (this contains the middle 50%). We want to compare where the middle 50% of the data is located in each group and whether there are any significant differences between the groups. The middle 50% of the data are the values that lie between the UQ and LQ. In other words, the IQR contains the middle 50% of the data. Has the box of one group shifted up compared to the other group? And is there any overlap?
Example
I notice that the middle 50% of female bag weights lies between 2000grams and 5000 grams. I notice that the middle 50% of male bag weights lies between 1600 grams and 5000 grams. I notice that there is female bag weight is shifted slightly to the right compared with male bag weights, and there is a lot of overlap in the weights of bags. Page9
5. Unusual features Outliers A member of a data set whose values for the variable in the data set are such that it lies well away from most of the other members of the data set. These must be VERY obvious.
Example
I notice that there may be an outlier with a weight of around 125kg.
Clusters Be cautious about commenting on small groupings.
Example
There are three clusters in the distribution; a group who did a very small amount or no paid work, a group who did part-time work (about 20 hours) and a group who did full-time work (about 35 to 40 hours). Page10
Example Compare and interpret the middle 50% of the data, and any unusual or interesting features.
Problem: Do the weights of girls’ school bags tend to be greater than the weights of boys’ school bags in the 14 middle schools?
I notice: that the female bag weights middle 50% appears to be shifted to the right compared to the males, and that there is a lot of overlap of data in the middle 50%. That there are no obvious outliers or clusters in this data set.
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Exercise Write a sentence comparing the middle 50% of the two groups, and any unusual or interesting features.. 1.
Do high school students in New Zealand who have a facebook account tend to have newer cellphones (months) than those who do not?
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2.
From high school students in New Zealand, is the average schoolbag weight (grams) carried by boys as they start secondary school more than the average weight carried by boys as they start intermediate school?
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Conclusion Random sample and Representativeness. Remember, that if we take a random sample (not a biased sample), we will have data that is representative of the population. This then means that we can have more confidence in our analysis and conclusion.
Accuracy Below is an image that you will see from the video. There are 3 different sample sizes shown, along with the original population. We can see that the bigger the sample size, the smaller the variation between samples, and therefore the increase in accuracy as your sample size increases.
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Exercise: 1) What happens to the accuracy as our sample size increases? Explain.
2) Which of these graphs has a smaller spread and is more accurate? Explain.
3) Complete the sentence: If I took a bigger sample …
Exercise: Problem. I wonder if Year 13's tend to send more text messages per day than Year ... Do you think everyone who doesn't have a phone would feel confident ... center best? The mean or the median? Why? 7) What about measures of spread. When should you use the range? When should you use the IQR? Explain.
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