ib math studies – topic 6
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IB Math Studies – Topic 6. Statistics. IB Course Guide Description. IB Course Guide Description. IB Course Guide Description. Categorical Data – Describes a particular quality or characteristic. It can be divided into categories. i.e. c olor of eyes or types of ice cream. - PowerPoint PPT PresentationTRANSCRIPT
IB Math Studies – Topic 6
Statistics
IB Course Guide Description
IB Course Guide Description
IB Course Guide Description
Types of Data
• Categorical Data – Describes a particular quality or characteristic. It can be divided into categories.
– i.e. color of eyes or types of ice cream
• Quantitative Data – Contains a numerical value. The information collected is termed numerical data.– Discrete – Takes exact number values and is often the result of counting.
• i.e. number of TVs or number of houses on a street– Continuous – Takes numerical values within a certain range and is often a
result of measuring.
• i.e. the height of seniors or the weight of freshman
Describing Data
Types of Distribution
Symmetric Distribution Positively Skewed Distribution
Negatively Skewed Distribution
24 families were surveyed to find the number of people in the family.
The results are:5, 9, 4, 4, 4, 5, 3, 4, 6, 8, 8, 5, 7, 6, 6, 8, 6, 9, 10, 7, 3, 5, 6, 6
a) Is this data discrete or continuous?
b) Construct a frequency table for the data.
c) Display the data using a column graph.
d) Describe the shape of the distribution. Are there any outliers?
e) What percentage of families have 5 or fewer people in them?
Example 1: Describing Data
Standard Deviation Formula
x
• x is any score• is the mean• n is the number of
scores
n
xx
2
Calculate the standard deviation
Values
2
4
5
5
6
6
7
35
• Calculate the mean• Subtract the mean from each
value• Square these• Add them• Divide by n• Take the square root
n
xx
2
xx 2xx
Standard Deviation on the GDC
xmean• Type data in List 1• 1-Var Stats L1
xdeviationstandard
• On paper you’ll see ‘s’ being used to standard for standard deviation. • But you should use the σ measurement from the calculator.
• The median is the second quartile, Q2
or 50th percentile• The lower quartile, Q1, is the median of the lower half of the
dataor 25th percentile
• The upper quartile, Q3, is the median of the upper half of the dataor 75th percentile
• The inter-quartile range is the difference in the upper quartile and the lower quartile.
IQR = Q3 – Q1
Measuring the Spread of Dara
Box Plots
• The inter-quartile range is the width of the box.
• The maximum length of each whisker is 1.5 times the inter-quartile range.
• Any data value that is larger than (or smaller than) 1.5 × IQR is marked as an outlier.
To Create a Box-and-Whisker Plot:
1) Make a number line.
2) Create the box between Q1 and Q3.
3) Draw in Q2.
4) Determine any outliers:
• Upper boundary = Q3 + 1.5(IQR)
• Lower boundary = Q1 – 1.5(IQR)
5) Plot any outliers.
6) Extend the whiskers to the maximum & minimum (provided they’re not outliers).
A hospital is trialing a new anesthetic drug and has collected data on how long the new and old drugs take before the patient becomes unconscious. They wish to
know which drug acts faster and which is more reliable.
Old drug times:8, 12, 9, 8, 16, 10, 14, 7, 5, 21, 13, 10, 8, 10 11, 8, 11, 9, 11, 14
New drug times:8, 12, 7, 8, 12, 11, 9, 8, 10, 8, 10, 9, 12, 8, 8, 7, 10, 7, 9, 9
Prepare a parallel box plot for the data sets and use it to compare the two drugs for speed and reliability.
Example :Box and Whisker Plots
FORMULA Pearson’s Correlation Coefficient: r
Correlation Coefficient on the GDC
• Turn on your Diagnostics
• Enter the data in L1 and L2
• LinReg L1, L2
In an experiment a vertical spring was fixed at its upper end. It was stretched by hanging different weights on its lower end. The length of the spring was then measured. The following readings were obtained.
Load (kg) x 0 1 2 3 4 5 6 7 8
Length (cm) y 23.5 25 26.5 27 28.5 31.5 34.5 36 37.5
It is given that the covariance Sxy is 12.17.
(d) (i) Write down the correlation coefficient, r, for these readings.
(ii) Comment on this result.
(b) (i) Write down the mean value of the load,
(ii) Write down the standard deviation of the load.
(iii) Write down the mean value of the length,
(iv) Write down the standard deviation of the length.
x
y
Example 1: Correlation Coefficient
Average speed in the metropolitan area and age of drivers
The r-value for this association is 0.027. Describe
the association.
Example 2: Correlation Coefficient
Drawing the Line of Best Fit
1. Calculate mean of x values , and mean of y values
2. Mark the mean point on the scatter plot
3. Draw a line through the mean point that is through the middle of the data– equal number of points above and below line
x y ,x y
Least Squares Regression Line
• Consider the set of points below.• Square the distances and find their
sum.• we want that sum to be small.• The regression line is used for
prediction purposes.• The regression line is less reliable
when extended far beyond the region of the data.
Line of Regression using GDC
• LinReg(ax +b) Test, L1, L2• where L1 contains your independent data.• and L2 contains your dependent data
The table shows the annual income and average weekly grocery bill for a selection of families
a) Construct a scatter plot to illustrate the data.b) Use technology to find the line of best fit.c) Estimate the weekly grocery bill for a family with an annual income of £95000.Comment on whether this estimate is likely to be reliable.
Example 3: Line of Regression
X2 Test of Independence
• The variables may be dependent:– Females may be
more likely to exercise regularly than males.
• The variables may be independent:– Gender has no
effect on whether they exercise regularly.
A chi-squared test is used to determine whether two variables
from the same sample are independent.
How to do it:
1) Write the null hypothesis (H0) and the alternate hypothesis (H1).
2) Create contingency tables for observed and expected values.
3) Calculate the chi-square statistic and degrees of freedom.
4) Find the chi-squared critical value (booklet).• Depends on the level of significance (p) and the
degrees of freedom (v).
5) Determine whether or not to accept the null hypothesis.
Contingency Tables
Observed Frequencies
Expected Frequencies
Column1 Column2 Totals
Row1 a b sum row1
Row2 c d sum row2
Totals Sum column1 Sum column2 total
Column1 Column2 Totals
Row1 sum row1
Row2 sum row2
Totals sum column 1 sum column 2 total
row1sum column1sum
total
row1sum column2sum
total
row2sum column1sum
total
row2sum column2sum
total
e
obscalc
f
ffX
2exp2
On the calculator:
Put your contingency table in matrix A STAT
TESTS C: χ2 Test
Observed: [A] Expected: [B] (this is where you want to go) Calculate
Output:Χ2 Χ2 calculated valuedf degrees of freedomin Matrix B expected values
Χ2 Statistic on the GDC
Find the Critical Value
• Get this from the formula booklet.
• Significance level (p) is always given in the problem.
A 5% significance level = 95% confidence level
• Degrees of freedom: v = (c - 1)(r – 1)
where c = number of columns in table
and r = number of rows in table
Accepting the Null Hypothesis
If X2calc < Critical Value
ACCEPT the null hypothesis
If X2calc > Critical Value
REJECT the null hypothesis
Important IB Notes:
• In examinations: the value of sxy will be given if required.
• sx represents the standard deviation of the variable X;
• sxy represents the covariance of the variables X and Y.
• A GDC can be used to calculate r when raw data is given.• For the EXAM students do NOT need to know how to
find the covariance.• But, for their project if they’re doing regression, then they
DO need to do covariance by hand so they can do the r by hand so they can get points for using a sophisticated math process.