The normal distribution and probability

Statistics, grade 11 (4 units) Lesson 11 of 20Learning

The normal distribution and probability

One pupil chosen at random

The area under the curve is also a probability: the part of the people in a range is the probability that a person chosen at random is in that range.

The marks in a test are normally distributed, with mean 71 and standard deviation 15.
One pupil is chosen at random.
What is the probability that the pupil scored more than 92?

$\displaystyle {z}={\frac{92-71}{15}}={1.4}$ , and $\displaystyle {\Phi(1.4)}={0.9192}$ .
$\displaystyle {P(X>92)}={1-0.9192}={0.0808}$ .

Two pupils: exactly one

Two pupils are chosen at random from a large number of pupils. What is the probability that exactly one of them scored more than 92?

The two choices hardly affect each other, so each pupil has the same probability, 0.0808. The tree has two stages, a stage for each pupil, and four paths.

Two paths fit: the first above 92 and the second not, or the other way round.
$\displaystyle {P}={0.0808\cdot0.9192+0.9192\cdot0.0808}={2\cdot0.0808\cdot0.9192\approx0.1485}$ .

Two independent tests

An institute has an entrance test in English and an entrance test in maths. Success in one test does not depend on success in the other.
In English the mean is 77, and the standard deviation 20.
In maths the mean is 81, and the standard deviation 10.
The probability that a pupil scored more than 90 in both tests is the product of the two probabilities:

English: $\displaystyle {z}={\frac{90-77}{20}}={0.65}$ , and $\displaystyle {P}={1-0.7422}={0.2578}$ .
Maths: $\displaystyle {z}={\frac{90-81}{10}}={0.9}$ , and $\displaystyle {P}={1-0.8159}={0.1841}$ .
Both: $\displaystyle 0.2578\cdot0.1841\approx0.0475$ .

Exercise 1

The delivery time of a pizza is normally distributed, with mean 30 minutes and standard deviation 8 minutes.
A delivery that takes more than 40 minutes counts as late.

a

What is the probability that a delivery chosen at random is late?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

A family ordered two pizzas, on two different days. Complete the tree, and work out the probability that exactly one delivery was late.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

What is the probability that at least one of the two deliveries was late?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Admission to a course requires a mark above 85 in two independent tests.
In the first test the mean is 70, and the standard deviation 12.
In the second test the mean is 75, and the standard deviation 8.
What is the probability that a candidate chosen at random was admitted?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

To get into a university, candidates must pass an entrance test, and the pass mark is 75.
This year two entrance tests were held.
80% of the candidates sat the first test, and the rest sat the second.
In both tests the marks were normally distributed.
In the first test the mean was 70, and the standard deviation 10.
In the second test the mean was 79, and the standard deviation 12.

a

A candidate who sat the first test is chosen at random. What is the probability that they passed?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

A candidate who sat the second test is chosen at random. What is the probability that they passed?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

A candidate is chosen at random. Complete the tree, and work out the probability that they passed the entrance test.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

d

A candidate chosen at random passed the entrance test. What is the probability that they sat the second test?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: statistics, grade 11 (4 units)

The mean and the standard deviation

The mean – $\displaystyle f_1, \dots, f_n$ the frequencies, $\displaystyle N$ the number of data
$\displaystyle \bar{x}=\frac{x_1f_1+x_2f_2+\dots+x_nf_n}{N}$
The standard deviation
$\displaystyle S=\sqrt{\frac{(x_1-\bar{x})^{2}f_1+(x_2-\bar{x})^{2}f_2+\dots+(x_n-\bar{x})^{2}f_n}{N}}$

The standard score

The standard score of the value $\displaystyle x$
$\displaystyle z=\frac{x-\bar{x}}{S}$
The value whose standard score is $\displaystyle z$
$\displaystyle x=\bar{x}+z\cdot S$

The normal distribution

The standard score in a normal distribution – $\displaystyle \mu$ the mean, $\displaystyle \sigma$ the standard deviation
$\displaystyle z=\frac{x-\mu}{\sigma}$
Within one standard deviation of the mean
$\displaystyle P(\mu-\sigma<X<\mu+\sigma)\approx68\%$
Within two standard deviations of the mean
$\displaystyle P(\mu-2\sigma<X<\mu+2\sigma)\approx95\%$
Above a standard score
$\displaystyle P(Z>z)=1-P(Z<z)$
The curve’s symmetry
$\displaystyle P(Z>z)=P(Z<-z)$
Between two standard scores
$\displaystyle P(z_1<Z<z_2)=P(Z<z_2)-P(Z<z_1)$

The cumulative standard normal table

In the table: the row – $\displaystyle z$ to its first decimal place; the column – the second. In a negative row the column goes further from 0: row $\displaystyle -1.2$ , column 3 holds $\displaystyle \Phi(-1.23)$ .
$\displaystyle \Phi(z)=P(Z<z)$

`z`	0	1	2	3	4	5	6	7	8	9
-3.0	0.0013	0.0013	0.0013	0.0012	0.0012	0.0011	0.0011	0.0011	0.0010	0.0010
-2.9	0.0019	0.0018	0.0018	0.0017	0.0016	0.0016	0.0015	0.0015	0.0014	0.0014
-2.8	0.0026	0.0025	0.0024	0.0023	0.0023	0.0022	0.0021	0.0021	0.0020	0.0019
-2.7	0.0035	0.0034	0.0033	0.0032	0.0031	0.0030	0.0029	0.0028	0.0027	0.0026
-2.6	0.0047	0.0045	0.0044	0.0043	0.0041	0.0040	0.0039	0.0038	0.0037	0.0036
-2.5	0.0062	0.0060	0.0059	0.0057	0.0055	0.0054	0.0052	0.0051	0.0049	0.0048
-2.4	0.0082	0.0080	0.0078	0.0075	0.0073	0.0071	0.0069	0.0068	0.0066	0.0064
-2.3	0.0107	0.0104	0.0102	0.0099	0.0096	0.0094	0.0091	0.0089	0.0087	0.0084
-2.2	0.0139	0.0136	0.0132	0.0129	0.0125	0.0122	0.0119	0.0116	0.0113	0.0110
-2.1	0.0179	0.0174	0.0170	0.0166	0.0162	0.0158	0.0154	0.0150	0.0146	0.0143
-2.0	0.0228	0.0222	0.0217	0.0212	0.0207	0.0202	0.0197	0.0192	0.0188	0.0183
-1.9	0.0287	0.0281	0.0274	0.0268	0.0262	0.0256	0.0250	0.0244	0.0239	0.0233
-1.8	0.0359	0.0351	0.0344	0.0336	0.0329	0.0322	0.0314	0.0307	0.0301	0.0294
-1.7	0.0446	0.0436	0.0427	0.0418	0.0409	0.0401	0.0392	0.0384	0.0375	0.0367
-1.6	0.0548	0.0537	0.0526	0.0516	0.0505	0.0495	0.0485	0.0475	0.0465	0.0455
-1.5	0.0668	0.0655	0.0643	0.0630	0.0618	0.0606	0.0594	0.0582	0.0571	0.0559
-1.4	0.0808	0.0793	0.0778	0.0764	0.0749	0.0735	0.0721	0.0708	0.0694	0.0681
-1.3	0.0968	0.0951	0.0934	0.0918	0.0901	0.0885	0.0869	0.0853	0.0838	0.0823
-1.2	0.1151	0.1131	0.1112	0.1093	0.1075	0.1056	0.1038	0.1020	0.1003	0.0985
-1.1	0.1357	0.1335	0.1314	0.1292	0.1271	0.1251	0.1230	0.1210	0.1190	0.1170
-1.0	0.1587	0.1562	0.1539	0.1515	0.1492	0.1469	0.1446	0.1423	0.1401	0.1379
-0.9	0.1841	0.1814	0.1788	0.1762	0.1736	0.1711	0.1685	0.1660	0.1635	0.1611
-0.8	0.2119	0.2090	0.2061	0.2033	0.2005	0.1977	0.1949	0.1922	0.1894	0.1867
-0.7	0.2420	0.2389	0.2358	0.2327	0.2296	0.2266	0.2236	0.2206	0.2177	0.2148
-0.6	0.2743	0.2709	0.2676	0.2643	0.2611	0.2578	0.2546	0.2514	0.2483	0.2451
-0.5	0.3085	0.3050	0.3015	0.2981	0.2946	0.2912	0.2877	0.2843	0.2810	0.2776
-0.4	0.3446	0.3409	0.3372	0.3336	0.3300	0.3264	0.3228	0.3192	0.3156	0.3121
-0.3	0.3821	0.3783	0.3745	0.3707	0.3669	0.3632	0.3594	0.3557	0.3520	0.3483
-0.2	0.4207	0.4168	0.4129	0.4090	0.4052	0.4013	0.3974	0.3936	0.3897	0.3859
-0.1	0.4602	0.4562	0.4522	0.4483	0.4443	0.4404	0.4364	0.4325	0.4286	0.4247
-0.0	0.5000	0.4960	0.4920	0.4880	0.4840	0.4801	0.4761	0.4721	0.4681	0.4641
0.0	0.5000	0.5040	0.5080	0.5120	0.5160	0.5199	0.5239	0.5279	0.5319	0.5359
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753
0.2	0.5793	0.5832	0.5871	0.5910	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.6480	0.6517
0.4	0.6554	0.6591	0.6628	0.6664	0.6700	0.6736	0.6772	0.6808	0.6844	0.6879
0.5	0.6915	0.6950	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.7190	0.7224
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549
0.7	0.7580	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852
0.8	0.7881	0.7910	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.8340	0.8365	0.8389
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.8770	0.8790	0.8810	0.8830
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.8980	0.8997	0.9015
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319
1.5	0.9332	0.9345	0.9357	0.9370	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.9750	0.9756	0.9761	0.9767
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817
2.1	0.9821	0.9826	0.9830	0.9834	0.9838	0.9842	0.9846	0.9850	0.9854	0.9857
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.9890
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916
2.4	0.9918	0.9920	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936
2.5	0.9938	0.9940	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.9960	0.9961	0.9962	0.9963	0.9964
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.9970	0.9971	0.9972	0.9973	0.9974
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.9980	0.9981
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986
3.0	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.9990	0.9990

The correlation coefficient

From raw data
$\displaystyle r=\frac{1}{N\cdot S_x\cdot S_y}\left[(x_1-\bar{x})(y_1-\bar{y})+\dots+(x_N-\bar{x})(y_N-\bar{y})\right]$
From standardised data – $\displaystyle z_x$ , $\displaystyle z_y$ the standard scores
$\displaystyle r=\frac{1}{N}\left[(z_x)_1(z_y)_1+\dots+(z_x)_N(z_y)_N\right]$
Always between $\displaystyle -1$ and $\displaystyle 1$
$\displaystyle -1\le r\le1$

The regression line

The slope of the regression line
$\displaystyle m=r\cdot\frac{S_y}{S_x}$
The regression line predicting $\displaystyle y$ from $\displaystyle x$ – through the point of the means
$\displaystyle y-\bar{y}=m(x-\bar{x})$