Questions - SPS SPS SM Statistics

Find the probability that she receives
1. exactly 2 Gold rewards,
2. at least 5 Gold rewards. In the next month Sam earns 300 rewards.
  She decides to use a Normal distribution to estimate the probability that she will receive at least 135 Gold rewards.
1. Find the mean and variance of this Normal distribution.
2. Estimate the probability that Sam will receive at least 135 Gold rewards.
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SPS SPS SM Statistics 2022 February Q14

14. Zac is planning to write a report on the music preferences of the students at his college. There is a large number of students at the college.

State one reason why Zac might wish to obtain information from a sample of students, rather than from all the students.
Amaya suggests that Zac should use a sample that is stratified by school year. Give one advantage of this method as compared with random sampling, in this context. Zac decides to take a random sample of 60 students from his college. He asks each student how many hours per week, on average, they spend listening to music during term. From his results he calculates the following statistics.
Mean
Standard
deviation
Median
Lower
quartile
Upper
quartile
21.0 4.20 20.5 18.0 22.9
Sundip tells Zac that, during term, she spends on average 30 hours per week listening to music. Discuss briefly whether this value should be considered an outlier.
Layla claims that, during term, each student spends on average 20 hours per week listening to music. Zac believes that the true figure is higher than 20 hours. He uses his results to carry out a hypothesis test at the $5 \%$ significance level. Assume that the time spent listening to music is normally distributed with standard deviation 4.20 hours. Carry out the test.
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SPS SPS SM Statistics 2022 January Q1

Each day Anna drives to work.

R is the event that it is raining.
$L$ is the event that Anna arrives at work late.

You are given that $\mathrm { P } ( R ) = 0.36 , \mathrm { P } ( L ) = 0.25$ and $\mathrm { P } ( R \cup L ) = 0.41$.
i. Draw a Venn diagram showing the events $R$ and $L$. Fill in the probability corresponding to each of the four regions of your diagram.
ii. Determine whether the events $R$ and $L$ are independent.
iii. Find $\mathrm { P } ( L \mid R )$

SPS SPS SM Statistics 2022 January Q2

2. The temperature of a supermarket fridge is regularly checked to ensure that it is working correctly. Over a period of three months the temperature (measured in degrees Celsius) is checked 600 times. These temperatures are displayed in the cumulative frequency diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{252d5094-6cdd-4379-bcd5-ca6a5cc48c7a-3_956_1497_1361_269}
i. Use the diagram to estimate the median and interquartile range of the data.
ii. Use your answers to part (i) to show that there are very few, if any, outliers in the sample. Below is the frequency table for these data:

Temperature

$( t$ degrees Celsius $)$

$3.0 \leq t \leq 3.4$

iii. Use the table to calculate estimates for the mean and standard deviation.
iv. The temperatures are converted from degrees Celsius to degrees Fahrenheit using the formula $F = 1.8 C + 32$. Hence estimate the mean and standard deviation of the temperatures in degrees Fahrenheit.

SPS SPS SM Statistics 2022 January Q3

3. The weights of Braeburn apples on display in a supermarket, measured in grams, are Normally distributed with mean 210.5 and standard deviation 15.2.
i. Find the probability that a randomly selected apple weighs at least 220 grams.
ii. 80 apples are selected at random.
a) Find the probability that more than 18 of these apples weigh at least 220 grams.
b) Find the expectation and standard deviation for the number of apples that weigh at least 220 grams.
c) State a suitable approximating distribution, including any parameters, for the number of apples that weigh at least 220 grams.
d) Explain why this approximating distribution is suitable. The supermarket also sells Cox's Orange Pippin apples. The weights of these apples, measured in grams, are Normally distributed with mean 185 and standard deviation $\sigma$.
iii. Given that $10 \%$ of randomly selected Cox's Orange Pippin apples weigh less than 170 grams, calculate the value of $\sigma$.

SPS SPS SM Statistics 2023 January Q1

Joseph drew a histogram to show information about one Local Authority. He used data from the "Age structure by LA 2011" tab in the large data set. The table shows an extract from the data that he used.
Age group 0 to 4
Frequency 2143
Joseph used a scale of $1 \mathrm {~cm} = 1000$ units on the frequency density axis. Calculate the height of the histogram block for the 0 to 4 class.
Magdalene wishes to draw a statistical diagram to illustrate some of the data from the "Method of travel by LA 2011" tab in the large data set. State why she cannot draw a histogram.

SPS SPS SM Statistics 2023 January Q2

2. Jane conducted a survey. She chose a sample of people from three towns, A, B and C. She noted the following information. 400 people were chosen.
230 people were adults.
55 adults were from town A .
65 children were from town A .
35 children were from town B .
150 people were from town $B$.

In the Printed Answer Booklet, complete the two-way frequency table.
\cline { 2 - 4 } \multicolumn{1}{c|}{} Town
\cline { 2 - 4 } \multicolumn{1}{c|}{} A B C Total
adult
child
Total
One of the people is chosen at random.
1. Find the probability that this person is an adult from town A .
2. Given that the person is from town A , find the probability that the person is an adult. For another survey, Jane wanted to choose a random sample from the 820 students living in a particular hostel. She numbered the students from 1 to 820 and then generated some random numbers on her calculator. The random numbers were 0.114287562 and 0.081859817 .
  Jane's friend Kareem used these figures to write down the following sample of five student numbers. 114, 142, 428, 287 and 756
  Jane used the same figures to write down the following sample of five student numbers.
  $114,287,562,81$ and 817
1. State, with a reason, which one of these samples is not random.
2. Explain why Jane omitted the number 859 from her sample.

SPS SPS SM Statistics 2023 January Q3

3. Pierre is a chef. He claims that $90 \%$ of his customers are satisfied with his cooking. Yvette suspects that Pierre is over-confident about the level of satisfaction amongst his customers. She talks to a random sample of 15 of Pierre's customers, and finds that 11 customers say that they are satisfied. She then performs a hypothesis test. Carry out the test at the 5\% significance level.

SPS SPS SM Statistics 2023 January Q4

The masses, in grams, of plums of a certain kind have the distribution $\mathrm { N } ( 55,18 )$. The heaviest $5 \%$ of plums are classified as extra large. Find the minimum mass of extra large plums.
The masses, in grams, of apples of a certain kind have the distribution $\mathrm { N } \left( 67 , \sigma ^ { 2 } \right)$. It is given that half of the apples have masses between 62 g and 72 g . Determine $\sigma$.

SPS SPS SM Statistics 2023 January Q5

5. Each member of a group of 27 people was timed when completing a puzzle.
The time taken, $x$ minutes, for each member of the group was recorded.
These times are summarised in the following box and whisker plot.
\includegraphics[max width=\textwidth, alt={}, center]{f03113c4-039e-4ead-9588-b4b83fb7eea9-08_381_1557_504_264}

Find the range of the times.
Find the interquartile range of the times. For these 27 people $\sum x = 607.5$ and $\sum x ^ { 2 } = 17623.25$
calculate the mean time taken to complete the puzzle,
calculate the standard deviation of the times taken to complete the puzzle. Taruni defines an outlier as a value more than 3 standard deviations above the mean.
State how many outliers Taruni would say there are in these data, giving a reason for your answer. Adam and Beth also completed the puzzle in $a$ minutes and $b$ minutes respectively, where $a > b$.
When their times are included with the data of the other 27 people
- the median time increases
- the mean time does not change
- Suggest a possible value for $a$ and a possible value for $b$, explaining how your values satisfy the above conditions.
- Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).

SPS SPS SM Statistics 2023 January Q6

6. The level, in grams per millilitre, of a pollutant at different locations in a certain river is denoted by the random variable $X$, where $X$ has the distribution $\mathrm { N } ( \mu , 0.0000409 )$. In the past the value of $\mu$ has been 0.0340 .
This year the mean level of the pollutant at 50 randomly chosen locations was found to be 0.0325 grams per millilitre. Test, at the $5 \%$ significance level, whether the mean level of pollutant has changed.

SPS SPS SM Statistics 2023 January Q7

7. A large college produces three magazines.
One magazine is about green issues, one is about equality and one is about sports. A student at the college is selected at random and the events $G , E$ and $S$ are defined as follows
$G$ is the event that the student reads the magazine about green issues $E$ is the event that the student reads the magazine about equality $S$ is the event that the student reads the magazine about sports The Venn diagram, where $p , q , r$ and $t$ are probabilities, gives the probability for each subset.
\includegraphics[max width=\textwidth, alt={}, center]{f03113c4-039e-4ead-9588-b4b83fb7eea9-12_533_903_790_548}

Find the proportion of students in the college who read exactly one of these magazines. No students read all three magazines and $\mathrm { P } ( G ) = 0.25$
Find
1. the value of $p$
2. the value of $q$ Given that $\mathrm { P } ( S \mid E ) = \frac { 5 } { 12 }$
find
1. the value of $r$
2. the value of $t$
Determine whether or not the events ( $S \cap E ^ { \prime }$ ) and $G$ are independent. Show your working clearly. END OF TEST

SPS SPS SM Statistics 2024 April Q1

1. The masses of a random sample of 120 boulders in a certain area were recorded. The results are summarized in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{d59e9fea-31cb-4b6d-b1d6-f09f912b5b37-04_773_1765_402_148}

Calculate the number of boulders with masses between 60 and 65 kg .
1. Use midpoints to find estimates of the mean and standard deviation of the masses of the boulders in the sample.
2. Explain why your answers are only estimates.
Use your answers to part (b)(i) to determine an estimate of the number of outliers, if any, in the distribution.
Give one advantage of using a histogram rather than a pie chart in this context.
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SPS SPS SM Statistics 2024 April Q2

11 marks

A certain five-sided die is biased with faces numbered 0 to 4 . The score, Y , on each throw is a random variable with probability distribution given by:
$Y$ 0 1 2 3 4
$\mathrm { P } ( Y = y )$ $a$ $b$ $c$ 0.1 0.15
where $a$, $b$ and $c$ are constants. $$\begin{aligned} & \mathrm { P } ( Y = 1 ) = \mathrm { P } ( Y \geq 3 )
& \mathrm { P } ( Y = 0 ) = \mathrm { P } ( Y = 2 ) - 0.1 \end{aligned}$$ Find the values of $a , b$ and $c$.
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The same die is thrown 10 times. Find the probability that there are not more than 4 throws on which the score is 3 , stating the distribution used as well as any modelling assumptions made.
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A game uses the same biased die. The die is thrown once. If it shows 1, 3 or 4 then this number is the final score. If it shows 0 or 2 then the die is thrown again and the final score is the sum of the numbers shown on the two throws.
(a) Find the probability that the final score is 3 .
(b) Given that the die is thrown twice, find the probability that the final score is 3 .
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SPS SPS SM Statistics 2024 April Q3

3. The table shows the increases, between 2001 and 2011, in the percentages of employees travelling to work by various methods, in the Local Authorities (LAs) in the North East region of the UK.

Geography code	Local authority	Work mainly at or from home	Underground, metro, light rail or tram	Bus, minibus or coach	Driving a car or van	Passenger in a car or van	On foot
E06000047	County Durham	0.74\%	0.05\%	-1.50\%	4.58\%	-2.99\%	-0.97\%
E06000005	Darlington	0.26\%	-0.01\%	-3.25\%	3.06\%	-1.28\%	0.99\%
E08000020	Gateshead	-0.01\%	-0.01\%	-2.28\%	4.62\%	-2.35\%	-0.18\%
E06000001	Hartlepool	0.03\%	-0.04\%	-1.62\%	4.80\%	-2.38\%	-0.26\%
E06000002	Middlesbrough	-0.34\%	-0.01\%	-2.32\%	2.19\%	-1.33\%	0.67\%
E08000021	Newcastle upon Tyne	0.10\%	-0.23\%	-0.67\%	-0.48\%	-1.51\%	1.75\%
E08000022	North Tyneside	0.05\%	0.54\%	-1.18\%	3.30\%	-2.21\%	-0.60\%
E06000048	Northumberland	1.39\%	-0.08\%	-0.95\%	3.50\%	-2.37\%	-1.44\%
E06000003	Redcar and Cleveland	-0.02\%	-0.01\%	-2.09\%	4.20\%	-2.06\%	-0.49\%
E08000023	South Tyneside	-0.36\%	2.03\%	-3.05\%	4.50\%	-2.41\%	-0.51\%
E06000004	Stockton-on-Tees	0.14\%	0.03\%	-2.02\%	3.52\%	-2.01\%	-0.15\%
E08000024	Sunderland	0.17\%	1.48\%	-3.11\%	4.89\%	-2.21\%	-0.52\%

\section*{Increase in percentage of employees travelling to work by various methods} The first two digits of the Geography code give the type of each of the LAs:
06: Unitary authority
07: Non-metropolitan district
08: Metropolitan borough

In what type of LA are the largest increases in percentages of people travelling by underground, metro, light rail or tram?
Identify two main changes in the pattern of travel to work in the North East region between 2001 and 2011. Now assume the following.
- The data refer to residents in the given LAs who are in the age range 20 to 65 at the time of each census.
- The number of people in the age range 20 to 65 who move into or out of each given LA, or who die, between 2001 and 2011 is negligible.
- Estimate the percentage of the people in the age range 20 to 65 in 2011 whose data appears in both 2001 and 2011.
- In the light of your answer to part (c), suggest a reason for the changes in the pattern of travel to work in the North East region between 2001 and 2011.
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SPS SPS SM Statistics 2024 April Q4

4. An online shopping company takes orders through its website. On average $80 \%$ of orders from the website are delivered within 24 hours. The quality controller selects 10 orders at random to check when they are delivered.

Find the probability that
(A) exactly 8 of these orders are delivered within 24 hours,
(B) at least 8 of these orders are delivered within 24 hours. The company changes its delivery method. The quality controller suspects that the changes will mean that fewer than $80 \%$ of orders will be delivered within 24 hours. A random sample of 18 orders is checked and it is found that 12 of them arrive within 24 hours.
Write down suitable hypotheses and carry out a test at the $5 \%$ significance level to determine whether there is any evidence to support the quality controller's suspicion.
A statistician argues that it is possible that the new method could result in either better or worse delivery times. Therefore it would be better to carry out a 2 -tail test at the $5 \%$ significance level. State the alternative hypothesis for this test. Assuming that the sample size is still 18, find the critical region for this test, showing all of your calculations.
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SPS SPS SM Statistics 2024 April Q5

5. In this question you must show detailed reasoning.
A disease that affects trees shows no visible evidence for the first few years after the tree is infected. A test has been developed to determine whether a particular tree has the disease. A positive result to the test suggests that the tree has the disease. However, the test is not $100 \%$ reliable, and a researcher uses the following model.

If the tree has the disease, the probability of a positive result is 0.95 .
If the tree does not have the disease, the probability of a positive result is 0.1 .
1. It is known that in a certain county, $A , 35 \%$ of the trees have the disease. A tree in county $A$ is chosen at random and is tested.

Given that the result is positive, determine the probability that this tree has the disease. A forestry company wants to determine what proportion of trees in another county, $B$, have the disease. They choose a large random sample of trees in county $B$. Each tree in the sample is tested and it is found that the result is positive for $43 \%$ of these trees.

By carrying out a calculation, determine an estimate of the proportion of trees in county $B$ that have the disease.
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SPS SPS SM Statistics 2024 January Q1

1. At the beginning of the academic year, all the pupils in year 12 at a college take part in an assessment. Summary statistics for the marks obtained by the 2021 cohort are given below.
$n = 205 \quad \sum x = 23042 \quad \sum x ^ { 2 } = 2591716$ Marks may only be whole numbers, but the Head of Mathematics believes that the distribution of marks may be modelled by a Normal distribution.

Calculate
- The mean mark
- The variance of the marks
- Use your answers to part (a) to write down a possible Normal model for the distribution of marks.

SPS SPS SM Statistics 2024 January Q2

2. The heights, in centimetres, of a random sample of 150 plants of a certain variety were measured. The results are summarised in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{0e73f1d0-5532-4995-b39e-759d82c2bd92-04_860_1684_367_130} One of the 150 plants is chosen at random, and its height, $X \mathrm {~cm}$, is noted.

Show that $\mathrm { P } ( 20 < X < 30 ) = 0.147$, correct to 3 significant figures. Sam suggests that the distribution of $X$ can be well modelled by the distribution $\mathrm { N } ( 40,100 )$.
1. Give a brief justification for the use of the normal distribution in this context.
2. Give a brief justification for the choice of the parameter values 40 and 100 .
Use Sam's model to find $\mathrm { P } ( 20 < X < 30 )$. Nina suggests a different model. She uses the midpoints of the classes to calculate estimates, $m$ and $s$, for the mean and standard deviation respectively, in centimetres, of the 150 heights. She then uses the distribution $\mathrm { N } \left( m , s ^ { 2 } \right)$ as her model.
Use Nina's model to find $\mathrm { P } ( 20 < X < 30 )$.

Complete the table in the Printed Answer Booklet to show the probabilities obtained from Sam's model and Nina's model.

By considering the different ranges of values of $X$ given in the table, discuss how well the two models fit the original distribution. Table for (e)(i):

$x$	Below 20	20 to 30	30 to 35	35 to 40	40 to 45	45 to 50	50 to 60	Above 60
Probability obtained from histogram	0.027	0.147	0.153	0.187	0.193	0.147	0.133	0.013
Probability obtained from Sam's model, N(40, 100)	0.023		0.150	0.191			0.136	0.023
Probability obtained from Nina's model, $\mathrm { N } \left( m , s ^ { 2 } \right)$	0.030		0.153	0.188			0.130	0.023

SPS SPS SM Statistics 2024 January Q3

3. Zac is planning to write a report on the music preferences of the students at his college. There is a large number of students at the college.

State one reason why Zac might wish to obtain information from a sample of students, rather than from all the students.
Amaya suggests that Zac should use a sample that is stratified by school year. Give one advantage of this method as compared with random sampling, in this context. Zac decides to take a random sample of 60 students from his college. He asks each student how many hours per week, on average, they spend listening to music during term. From his results he calculates the following statistics.
Mean
Standard
deviation
Median
Lower
quartile
Upper
quartile
21.0 4.20 20.5 18.0 22.9
Sundip tells Zac that, during term, she spends on average 30 hours per week listening to music. Discuss briefly whether this value should be considered an outlier.
Layla claims that, during term, each student spends on average 20 hours per week listening to music. Zac believes that the true figure is higher than 20 hours. He uses his results to carry out a hypothesis test at the $5 \%$ significance level. Assume that the time spent listening to music is normally distributed with standard deviation 4.20 hours. Carry out the test.

SPS SPS SM Statistics 2024 January Q4

4. The table shows the increases, between 2001 and 2011, in the percentages of employees travelling to work by various methods, in the Local Authorities (LAs) in the North East region of the UK.

Geography code	Local authority	Work mainly at or from home	Underground, metro, light rail or tram	Bus, minibus or coach	Driving a car or van	Passenger in a car or van	On foot
E06000047	County Durham	0.74\%	0.05\%	-1.50\%	4.58\%	-2.99\%	-0.97\%
E06000005	Darlington	0.26\%	-0.01\%	-3.25\%	3.06\%	-1.28\%	0.99\%
E08000020	Gateshead	-0.01\%	-0.01\%	-2.28\%	4.62\%	-2.35\%	-0.18\%
E06000001	Hartlepool	0.03\%	-0.04\%	-1.62\%	4.80\%	-2.38\%	-0.26\%
E06000002	Middlesbrough	-0.34\%	-0.01\%	-2.32\%	2.19\%	-1.33\%	0.67\%
E08000021	Newcastle upon Tyne	0.10\%	-0.23\%	-0.67\%	-0.48\%	-1.51\%	1.75\%
E08000022	North Tyneside	0.05\%	0.54\%	-1.18\%	3.30\%	-2.21\%	-0.60\%
E06000048	Northumberland	1.39\%	-0.08\%	-0.95\%	3.50\%	-2.37\%	-1.44\%
E06000003	Redcar and Cleveland	-0.02\%	-0.01\%	-2.09\%	4.20\%	-2.06\%	-0.49\%
E08000023	South Tyneside	-0.36\%	2.03\%	-3.05\%	4.50\%	-2.41\%	-0.51\%
E06000004	Stockton-on-Tees	0.14\%	0.03\%	-2.02\%	3.52\%	-2.01\%	-0.15\%
E08000024	Sunderland	0.17\%	1.48\%	-3.11\%	4.89\%	-2.21\%	-0.52\%

In what type of LA are the largest increases in percentages of people travelling by underground, metro, light rail or tram?
Identify two main changes in the pattern of travel to work in the North East region between 2001 and 2011. Now assume the following.
- The data refer to residents in the given LAs who are in the age range 20 to 65 at the time of each census.
- The number of people in the age range 20 to 65 who move into or out of each given LA, or who die, between 2001 and 2011 is negligible.
- Estimate the percentage of the people in the age range 20 to 65 in 2011 whose data appears in both 2001 and 2011.
- In the light of your answer to part (c), suggest a reason for the changes in the pattern of travel to work in the North East region between 2001 and 2011.

SPS SPS SM Statistics 2024 January Q5

5. Labrador puppies may be black, yellow or chocolate in colour. Some information about a litter of 9 puppies is given in the table.

	male	female
black	1	3
yellow	2	1
chocolate	1	1

Four puppies are chosen at random to train as guide dogs.
(b) Determine the probability that at least 3 black puppies are chosen.
(c) Determine the probability that exactly 3 females are chosen given that at least 3 black puppies are chosen.
(d) Explain whether the 2 events
'choosing exactly 3 females' and 'choosing at least 3 black puppies' are independent events. A firm claims that no more than $2 \%$ of their packets of sugar are underweight. A market researcher believes that the actual proportion is greater than $2 \%$. In order to test the firm's claim, the researcher weighs a random sample of 600 packets and carries out a hypothesis test, at the $5 \%$ significance level, using the null hypothesis $p = 0.02$.
(a) Given that the researcher's null hypothesis is correct, determine the probability that the researcher will conclude that the firm's claim is incorrect.
(b) The researcher finds that 18 out of the 600 packets are underweight. A colleague says
" 18 out of 600 is $3 \%$, so there is evidence that the actual proportion of underweight bags is greater than $2 \%$." Criticise this statement.

SPS SPS SM Statistics 2024 January Q7

7. The probability distribution of a random variable $X$ is modelled as follows.
$\mathrm { P } ( X = x ) = \begin{cases} \frac { k } { x } & x = 1,2,3,4 ,
0 & \text { otherwise, } \end{cases}$
where $k$ is a constant.

Show that $k = \frac { 12 } { 25 }$.
Show in a table the values of $X$ and their probabilities.
The values of three independent observations of $X$ are denoted by $X _ { 1 } , X _ { 2 }$ and $X _ { 3 }$. Find $\mathrm { P } \left( X _ { 1 } > X _ { 2 } + X _ { 3 } \right)$. In a game, a player notes the values of successive independent observations of $X$ and keeps a running total. The aim of the game is to reach a total of exactly 7 .
Determine the probability that a total of exactly 7 is first reached on the 5th observation. END OF TEST

SPS SPS SM Statistics 2025 April Q1

It is known that, under standard conditions, $12 \%$ of light bulbs from a certain manufacturer have a defect. A quality improvement process has been implemented, and a random sample of 200 light bulbs produced after the improvements was selected. It was found that 15 of the 200 light bulbs were defective.
1. State one assumption needed in order to use a binomial model for the number of defective light bulbs in the sample.
2. Test, at the $5 \%$ significance level, whether the proportion of defective light bulbs has decreased under the new process.
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The histogram shows information about the lengths, $l$ centimetres, of a sample of worms of a certain species.
\includegraphics[max width=\textwidth, alt={}, center]{a18b06b1-053e-45b2-9c28-2f125cf6cbba-06_805_1151_269_280} The number of worms in the sample with lengths in the class $3 \leqslant l < 4$ is 30 .
Find the number of worms in the sample with lengths in the class $0 \leqslant l < 2$.
Find an estimate of the number of worms in the sample with lengths in the range $4.5 \leqslant l < 5.5$.
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SPS SPS SM Statistics 2025 April Q3

1 marks

3. A researcher has collected data on the heights of a sample of adults but has encoded the actual values using a linear transformation of the form $a X + b$, where $X$ represents the original height in centimetres.
Given the following information about the encoded data:
The mean of the encoded heights is 5.4 cm
The standard deviation of the encoded heights is 2.0 cm
The researcher knows that the transformation used was $0.2 X - 30$

Find the mean of the original heights in the sample.
Find the standard deviation of the original heights in the sample.
If an encoded height value is 6.8 , what was the original height in centimetres?
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\(Y\)	0	1	2	3	4
\(\mathrm { P } ( Y = y )\)	\(a\)	\(b\)	\(c\)	0.1	0.15

\cline { 2 - 4 } \multicolumn{1}{c\|}{}	Town
\cline { 2 - 4 } \multicolumn{1}{c\|}{}	A	B	C	Total
adult
child
Total

\(x\)	Below 20	20 to 30	30 to 35	35 to 40	40 to 45	45 to 50	50 to 60	Above 60
Probability obtained from histogram	0.027	0.147	0.153	0.187	0.193	0.147	0.133	0.013
Probability obtained from Sam's model, N(40, 100)	0.023		0.150	0.191			0.136	0.023
Probability obtained from Nina's model, \(\mathrm { N } \left( m , s ^ { 2 } \right)\)	0.030		0.153	0.188			0.130	0.023

Age group	0 to 4
Frequency	2143

Questions — SPS SPS SM Statistics (77 questions)

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SPS SPS SM Statistics 2024 April Q1

SPS SPS SM Statistics 2024 April Q2

SPS SPS SM Statistics 2024 April Q3

SPS SPS SM Statistics 2024 April Q4

SPS SPS SM Statistics 2024 April Q5

SPS SPS SM Statistics 2024 January Q1

SPS SPS SM Statistics 2024 January Q2

SPS SPS SM Statistics 2024 January Q3

SPS SPS SM Statistics 2024 January Q4

SPS SPS SM Statistics 2024 January Q5

SPS SPS SM Statistics 2024 January Q7

SPS SPS SM Statistics 2025 April Q1

SPS SPS SM Statistics 2025 April Q3