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core maths > paper 2 > Flashcards

paper 2 Flashcards

1
Q

NORMAL DISTRIBUTION

What are the features of a normal distribution curve?

A
  • a bell shape curve
  • a single peak
  • symmetrical about the mean
    • 50% above and 50% below the data
  • most of the data is within 1 s.d. of the mean
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2
Q

What proportions of the sample is at which point?

A

68% = within 1 s.d. of the mean (µ + 1∂ and µ - 1∂)

95% = within 2 s.d. of the mean (µ + 2∂ and µ - 2∂)

99.7% = within 3 s.d. of the mean (µ + 3∂ and µ - 3∂)

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3
Q

What are the notations for normal distribution?

A

µ = mean

∂ = standard deviation

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4
Q

What is the notation for a random X that is normally distributed?

A

X ~ N (µ,∂)

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5
Q

What does ∂² mean?

A

∂² is the standard deviation squared and is called VARIANCE

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6
Q

What are the rules for probability of normal distribution?

A

If X<15 LEAVE IT
(if it is pointing at number (less than))

If X>15 subtract it from 1
(pointing at X (greater than))

  • if the answer is negative then DO THE OPPOSITE
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7
Q

How do you calculate the probability of normal distribution on a calculator?

A
  1. Press the MENU button and press 7
  2. Press ‘2: Normal CD’
  3. If X>15 (less than) put:
    LOWER = -10000000
    UPPER = 15 (or whatever number it is)

If X<15 (greater than) put:
LOWER = 15
UPPER = 10000000

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8
Q

What are some other points about finding ND on the calculator?

A
  • calculator value is ALWAYS LESS THAN

- the same rules still apply about whether to subtract from zero or not

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9
Q

INVERSE DISTRIBUTION

What is the inverse normal?

A

INVERSE NORMAL = Area = probability/percentile

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9
Q

INVERSE DISTRIBUTION

What is the inverse normal?

A

INVERSE NORMAL = Area = probability/percentile
(e.g. 95% = 0.95
Area = 0.95 )

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10
Q

How do you find inverse normal on a calculator?

A
  1. Press the MENU button and press 7
  2. Press ‘3: Inverse Normal’
  3. Then input the area (to the left of the boundary), the standard deviation and the mean
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11
Q

EXAMPLE QUESTION OF INVERSE NORMAL

X~N (25,4)

  1. Find ‘a’ given that P(X = 0.27
A
  1. Input the information into the calculator:
    Area :0.27
    ∂ :2
    µ. :25
    XInv = 23.77
  2. write as a=23.77
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12
Q

EXAMPLE QUESTION 2 OF INVERSE DISTRIBUTION

X~N (25,4)
P(24

A 
1. find P(X<24) with NORMAL distribution
   LOWER: -100000
   UPPER: 24
   ∂: 2
   µ: 25 
                              P=0.30854 (the area on the left of the 24 boundary)
  1. Find P(X
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13
Q

CONFIDENCE INTERVALS

What will CONFIDENCE be based on?

A

THE SIZE OF THE SAMPLE = the larger the size of the sample, the closer the estimate is likely to be to the true population mean

THE VARIANCE = If readings are generally more varied then the estimate will be less reliable

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14
Q

How do you you calculate the standard error? what is it?

A

Standard error = ∂/√n

Standard error is how different the population mean is likely to be from a sample mean

(How different the population mean is from the point estimate)

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15
Q

What is the formula for confidence intervals?

A

x̅ ± 1.96 ∂/√n

(with µ in middle)

x̅ = sample mean
n = sample size 
∂ = population standard deviation
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16
Q

How does this formula look written out in full?

A

x̅ - 1.96 ∂/√n < µ < x̅ + 1.96 ∂/√n = 95%

(the numbers will change based on your level of confidence)

- = lower confidence limit 
\+ = upper confidence limit
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17
Q

What are the decimal numbers that are substituted into the formula for different confidence intervals?

A 
90% = 1.64
95% = 1.96
98% = 2.33
99% = 2.57
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18
Q

EXAMPLE CONFIDENCE INTERVALS QUESTION

A sample of 16 fish with a mean length in the sample of 28cm. The standard deviation of this length is 4cm. Show a 95% confidence interval for the mean length of the fish in the length.

A
  1. x̅ = 28
    n = 16
    ∂ = 4
    CI = 95%
  2. UPPER = 28 + 1.96 (4/√16)
    = 29.96
    LOWER = 28 - 1.96 (4/√16)
    = 26.04
  3. Confidence interval = 26.04 < µ < 29.96
19
Q

What does PMCC stand for?

A

Product moment correlation coefficient

20
Q

How is the PMCC notated?

A

It is usually notated with the letter ‘r’

21
Q

What is the letter ‘r’ (PMCC)

A
  • r is a number between -1 and 1
    (- 1< r < 1)
\+1 = perfect positive correlation 
-1 = perfect negative correlation 
0 = no correlation
22
Q

How do you calculate the PMCC on the calculator?

A
  1. Press the MENU button and press 6 (statistics)
  2. Press ‘2: a+bx’
  3. input all the x and y data points into the table
  4. press option (OPTN)
  5. press ‘4: regression calculation)
  6. use ‘r’ for the PMCC
23
Q

can the ‘r’ value be affected by outliers?

A

yes it can

24
Q

What is the equation for the regression line?

A

y = a + bx

a = y - intercept 
b = gradient

(substitute the letters from the question into the formula swell as the numbers e.g. if the letters were w and l the equation would be W = a + bl)

25
Q

How do you calculate regression line of the calculator?

A
  1. Press the MENU button and press 6 (statistics)
  2. Press ‘2: a+bx’
  3. input all the x and y data points into the table
  4. press option (OPTN)
  5. press ‘4: regression calculation)
  6. use ‘a’ and ‘b’ for regression line
26
Q

What do you need to do when answering the question?

A
  1. write the a and the b value
  2. substitute these numbers into the formula
  3. then answer the question by drawing the line or explaining what it shows
27
Q

MEAN AND STANDARD DEVIATION

How is mean represented and worked out with listed data and frequency?

A

x̅ = ∑fx / ∑f

x = individual data entries 
f = frequency
28
Q

How is mean represented and worked out with grouped data and frequency?

A

x̅ = ∑fx / ∑f

x = grouped data MIDPOINTS
f = frequency
29
Q

What is the advantage of the mean?

A

It is the most used average and uses every item of data.

30
Q

What is the disadvantage of the mean?

A

It might not be representative if there is an extreme value (affected by outliers)

31
Q

what is standard deviation?

A
  • A measure of SPREAD that uses all of the data

- a HIGHER s.d. means that the data is MORE SPREAD OUT (and the opposite if it is low)

32
Q

What is the advantage of using standard deviation?

A

It uses all of the data

33
Q

What is the disadvantage of using standard deviation?

A

It takes longer to calculate and is therefore time consuming

34
Q

How do you calculate the standard deviation of a set of LISTED data?

A
  1. find the mean of the data
  2. Square all of the values SEPARATELY then add them together
  3. use the formula:

√∑x̅i²/n - x̅²

n = the number of values 
x̅² = mean squared
  1. get s.d.
35
Q

How to find the standard deviation of grouped data?

A
  1. find the mean of the data
  2. find the MIDPOINTs of the group
  3. multiply midpoints by the FREQUENCY
  4. add all of the values up
  5. use formula:

√∑fx²/∑f - x̅²

∑fx² = value from above
∑f = sum of frequency 
x̅² = mean squared
36
Q

What is the variance?

A

Standard deviation squared (∂²)

37
Q

How do you calculate standard deviation on a calculator? (listed data)

A
  1. Press the MENU button and press 6 (statistics)
  2. Press ‘1: 1-variable’
  3. Then press ‘SHIFT’ ‘MENU’, go down a page and press ‘3: statistics’
  4. press (2 : OFF)
  5. input your data
  6. then press option (OPTN)
  7. then press 3: 1-variable calc’
  8. find ∂x for standard deviation
38
Q

How do you calculate standard deviation on a calculator? (grouped data)

A
  1. Press the MENU button and press 6 (statistics)
  2. Press ‘1: 1-variable’
  3. Then press ‘SHIFT’ ‘MENU’, go down a page and press ‘3: statistics’
  4. press (1 : ON)
  5. input your data (for x input the MIDPOINTS and enter the frequencies)
  6. then press option (OPTN)
  7. then press 3: 1-variable calc’
  8. find ∂x for standard deviation
39
Q

What factors do you need to look out for when doing critical analysis?

A
  • Is there any data to back up statements made?
  • Use of vague or emotive language.
  • Has the writer assumed too much either about the subject matter or the readers knowledge?
  • how is the sample size and is it proportional to the research that they are doing?
  • (if a graph) does it have axis/ are the axis misleading?
  • Is it showing what it is meant to?
  • Is there errors in the data?
  • Is it even possible?
  • Are the scales distorting the data?
  • Is it the best type of graph?
40
Q

What is the rule for outliers?

A

AN OUTLIER = an extreme value

  • it is generally when we’re 1.5 IQRs beyond the lower and upper quantities
41
Q

What is an example of an outlier question?

A 
IQR = 7
UQ = 22
LQ = 15

7 x 1.5 = 10.5
= 22 + 10.5 = 32.5
= 15 - 10.5 = 4.5

42
Q

What is a point estimate?

A
  • the process of finding an approximate value of some parameter—such as the mean (average)—of a population from random samples of the population.
  • point estimation involves the use of sample data to calculate a single value which is to serve as a “best guess” or “best estimate” of an unknown population parameter. (e.g. finding the mean)
  • knowing that the mean of a sample is called a ‘point estimate’ for the mean of the population
43
Q

How do you calculate point estimate?

A
  • A point estimate of the mean of a population is determined by calculating the mean of a sample drawn from the population.
  • The calculation of the mean is the sum of all sample values divided by the number of values.
44
Q

How do you increase the accuracy of a point estimate?

A

The accuracy of the point estimate is likely to be improved by increasing the sample size

45
Q

what is the equation for standardising?

A

= first find the area from the numbers (e.g. 0.45)

∂

N = number in probability

= you then look at the statistical tables

core maths (2 decks)

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