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Are our results reliable enough to support a conclusion?

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Imagine we chose two children at random from two class rooms… D8C1 … and compare their height …

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D8C1 … we find that one pupil is taller than the other WHY?

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REASON 1: There is a significant difference between the two groups, so pupils in C1 are taller than pupils in D8 D8 YEAR 7 C1 YEAR 11

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REASON 2: By chance, we picked a short pupil from D8 and a tall one from C1 D8C1 TITCH (Year 9) HAGRID (Year 9)

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How do we decide which reason is most likely? MEASURE MORE STUDENTS!!!

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If there is a significant difference between the two groups… D8C1 … the average or mean height of the two groups should be very… … DIFFERENT

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If there is no significant difference between the two groups… D8C1 … the average or mean height of the two groups should be very… … SIMILAR

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Remember: Living things normally show a lot of variation, so…

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It is VERY unlikely that the mean height of our two samples will be exactly the same C1 Sample Average height = 162 cm D8 Sample Average height = 168 cm Is the difference in average height of the samples large enough to be significant?

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We can analyse the spread of the heights of the students in the samples by drawing histograms Here, the ranges of the two samples have a small overlap, so… … the difference between the means of the two samples IS probably significant. 2 4 6 8 10 12 14 16 Frequency 140- 149 150- 159 160- 169 170- 179 180- 189 Height (cm) C1 Sample 2 4 6 8 10 12 14 16 Frequency 140- 149 150- 159 160- 169 170- 179 180- 189 Height (cm) D8 Sample

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Here, the ranges of the two samples have a large overlap, so… … the difference between the two samples may NOT be significant. The difference in means is possibly due to random sampling error 2 4 6 8 10 12 14 16 Frequency 140- 149 150- 159 160- 169 170- 179 180- 189 Height (cm) C1 Sample 2 4 6 8 10 12 14 16 Frequency 140- 149 150- 159 160- 169 170- 179 180- 189 Height (cm) D8 Sample

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To decide if there is a significant difference between two samples we must compare the mean height for each sample… … and the spread of heights in each sample. Statisticians calculate the standard deviation of a sample as a measure of the spread of a sample S x = Σx 2 - (Σx) 2 n n - 1 Where: Sx is the standard deviation of sample Σ stands for ‘sum of’ x stands for the individual measurements in the sample n is the number of individuals in the sample You can calculate standard deviation using the formula:

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Student’s t-test The Student’s t-test compares the averages and standard deviations of two samples to see if there is a significant difference between them. We start by calculating a number, t t can be calculated using the equation: ( x 1 – x 2 ) (s 1 ) 2 n1n1 (s 2 ) 2 n2n2 + t = Where: x 1 is the mean of sample 1 s 1 is the standard deviation of sample 1 n 1 is the number of individuals in sample 1 x 2 is the mean of sample 2 s 2 is the standard deviation of sample 2 n 2 is the number of individuals in sample 2

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Worked Example: Random samples were taken of pupils in C1 and D8 Their recorded heights are shown below… Students in C1Students in D8 Student Height (cm) 145149152153154148153157161162 154158160166 162163167172 166167175177182175177183185187 Step 1: Work out the mean height for each sample 161.60C1: x 1 =168.27D8: x 2 = Step 2: Work out the difference in means 6.67x 2 – x 1 = 168.27 – 161.60 =

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Step 3: Work out the standard deviation for each sample C1: s 1 =10.86D8: s 2 =11.74 Step 4: Calculate s 2 /n for each sample (s 1 ) 2 n1n1 = C1: 10.86 2 ÷ 15 =7.86 (s 2 ) 2 n2n2 = D8: 11.74 2 ÷ 15 = 9.19

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Step 5: Calculate (s 1 ) 2 n1n1 + (s 2 ) 2 n2n2 (s 1 ) 2 n1n1 + (s 2 ) 2 n2n2 = (7.86 + 9.19)=4.13 Step 6: Calculate t(Step 2 divided by Step 5) t = (s 1 ) 2 n1n1 + (s 2 ) 2 n2n2 = x 2 – x 1 6.67 4.13 = 1.62

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Step 7: Work out the number of degrees of freedom d.f. = n 1 + n 2 – 2 =15 + 15 – 2 =28 Step 8: Find the critical value of t for the relevant number of degrees of freedom Use the 95% (p=0.05) confidence limit Critical value = 2.048 Our calculated value of t is below the critical value for 28d.f., therefore, there is no significant difference between the height of students in samples from C1 and D8

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A one- or two-tailed t-test is determined by whether the total area of a is placed in one tail or divided equally between the two tails. The one- tailed t-test is performed if the results are interesting only if they turn out in a particular direction.

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The two-tailed t-test is performed if the results would be interesting in either direction. The choice of a one- or two-tailed t-test effects the hypothesis testing procedure in a number of different ways.

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