5.06c Fit other distributions: discrete and continuous

5. Kylie used video technology to monitor the direction of flight, as a bearing, \(x\) degrees, for 450 honeybees that left her beehive during a particular morning. Kylie's results are summarised in the table below. Kylie believes that a continuous uniform distribution over the interval [0,360] is a suitable model for the direction of flight. Stating your hypotheses clearly, use a 1\% level of significance to test Kylie's belief. Show your working clearly.

7 The following table shows observed frequencies, where \(x\) is an integer, from an experiment to test whether or not a six-sided die is biased. A goodness of fit test is conducted to determine if there is evidence that the die is biased.

1. Write down suitable null and alternative hypotheses for this test. It is found that the null hypothesis is not rejected at the \(5 \%\) significance level.
2. Hence
   1. find the minimum value of \(x\)
   2. determine the minimum number of times the die was rolled.

1. It is suggested that the delay, in hours, of certain flights from a particular country may be modelled by the continuous random variable, \(T\), with probability density function

$$f ( t ) = \left\{ \begin{array} { c l } \frac { 2 } { 25 } t & 0 \leqslant t < 5 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that for \(0 \leqslant a \leqslant 4\) $$P ( a \leqslant T < a + 1 ) = \frac { 1 } { 25 } ( 2 a + 1 )$$ A random sample of 150 of these flights is taken. The delays are summarised in the table below.

   Delay ( \(\boldsymbol { t }\) hours)	Frequency
   \(0 \leqslant t < 1\)	10
   \(1 \leqslant t < 2\)	13
   \(2 \leqslant t < 3\)	24
   \(3 \leqslant t < 4\)	35
   \(4 \leqslant t < 5\)	68

2. Test, at the \(5 \%\) significance level, whether the given probability density function is a suitable model for these delays.
   You should state your hypotheses, expected frequencies, test statistic and the critical value used.

1. The manager of a company making ice cream believes that the proportions of people in the population who prefer vanilla, chocolate, strawberry and other are in the ratio \(10 : 5 : 2 : 3\)

The manager takes a random sample of 400 customers and records their age and favourite ice cream flavour. The results are shown in the table below.

1. Use the data in the table to test, at the \(5 \%\) level of significance, the manager's belief. You should state your hypotheses, test statistic, critical value and conclusion clearly. A researcher wants to investigate whether or not there is a relationship between the age of a customer and their favourite ice cream flavour. In order to test whether favourite ice cream flavour and age are related, the researcher plans to carry out a \(\chi ^ { 2 }\) test.
2. Use the table to calculate expected frequencies for the group
   1. teenagers whose favourite ice cream flavour is vanilla,
   2. adults whose favourite ice cream flavour is chocolate.
3. Write down the number of degrees of freedom for this \(\chi ^ { 2 }\) test.

4. Luka wants to carry out a survey of students at his school. He obtains a list of all 280 students.

1. Explain how he can use this list to select a systematic sample of 40 students. Luka is trying to make his own random number table. He generates 400 digits to put in his table. Figure 1 shows the frequency of each digit in his table. \begin{table}[h]

   Digit generated	0	1	2	3	4	5	6	7	8	9
   Frequency	36	42	33	41	44	43	48	38	32	43

   \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{table} A test is carried out at the \(10 \%\) level of significance to see if the digits Luka generates follow a uniform distribution. For this test \(\sum \frac { ( \mathrm { O } - \mathrm { E } ) ^ { 2 } } { \mathrm { E } } = 5.9\)
2. Determine the conclusion of this test.
   (3) The digits generated by Luka are taken two at a time to form two-digit numbers. Figure 2 shows the frequency of two-digit numbers in his table. \begin{table}[h]

   Two-digit numbers generated	\(00 - 19\)	\(20 - 39\)	\(40 - 59\)	\(60 - 79\)	\(80 - 99\)
   Frequency	31	49	30	42	48

   \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{table}
3. Test, at the \(10 \%\) level of significance, whether the two-digit numbers generated by Luka follow a uniform distribution. You should state the hypotheses, the degrees of freedom and the critical value used for this test. There are 70 students in Year 12 at his school.
4. State, giving a reason, the advice you would give to Luka regarding the use of his table of numbers for generating a simple random sample of 10 of the Year 12 students.

2. Andy has some apple trees. Over many years she has graded each apple from her trees as \(A , B , C , D\) or \(E\) according to the quality of the apple, with \(A\) being the highest quality and \(E\) being the lowest quality. She knows that the proportion of apples in each grade produced by her trees is as follows. Raj advises Andy to add potassium to the soil around her apple trees. Andy believes that adding potassium will not affect the distribution of grades for the quality of the apples. To test her belief Andy adds potassium to the soil around her apple trees. The following year she counts the number of apples in each grade. The number of apples in each grade is shown in the table below. Test Andy's belief using a \(5 \%\) level of significance. Show your working clearly, stating your hypotheses, expected frequencies and degrees of freedom. 2 continued

6. A total of 228 items are collected from an archaeological site. The distance from the centre of the site is recorded for each item. The results are summarised in the table below. Test, at the \(5 \%\) level of significance, whether or not the data can be modelled by a continuous uniform distribution. State your hypotheses clearly.

4. Customers at a post office are timed to see how long they wait until being served at the counter. A random sample of 50 customers is chosen and their waiting times, \(x\) minutes, are summarised in Table 1. \begin{table}[h] \end{table}

1. Show that an estimate of \(\bar { x } = 5.49\) and an estimate of \(s _ { x } ^ { 2 } = 6.88\) The post office manager believes that the customers' waiting times can be modelled by a normal distribution.
   Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2. \begin{table}[h]

   Waiting Time	\(x < 3\)	\(3 - 5\)	\(5 - 6\)	\(6 - 8\)	\(x > 8\)
   Expected Frequency	8.56	12.73	7.56	\(a\)	\(b\)

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
2. Find the value of \(a\) and the value of \(b\).
3. Test, at the \(5 \%\) level of significance, the manager's belief. State your hypotheses clearly.

6. Bags of \(\pounds 1\) coins are paid into a bank. Each bag contains 20 coins. The bank manager believes that \(5 \%\) of the \(\pounds 1\) coins paid into the bank are fakes. He decides to use the distribution \(X \sim \mathrm {~B} ( 20,0.05 )\) to model the random variable \(X\), the number of fake \(\pounds 1\) coins in each bag.

1. State the assumptions necessary for the binomial distribution to be an appropriate model in this case. The bank manager checks a random sample of 150 bags of \(\pounds 1\) coins and records the number of fake coins found in each bag. His results are summarised in Table 1. \begin{table}[h]

   Number of fake coins in each bag	0	1	2	3	4 or more
   Observed frequency	43	62	26	13	6
   Expected frequency	53.8	56.6	\(r\)	8.9	\(s\)

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. Calculate the values of \(r\) and \(s\), giving your answers to 1 decimal place.
3. Carry out a hypothesis test, at the \(5 \%\) significance level, to see if the data supports the bank manager's statistical model. State your hypotheses clearly. Question 6 parts (d) and (e) are continued on page 24 The assistant manager thinks that a binomial distribution is a good model but suggests that the proportion of fake coins is higher than \(5 \%\). She calculates the actual proportion of fake coins in the sample and uses this value to carry out a new hypothesis test on the data. Her expected frequencies are shown in Table 2. \begin{table}[h]

   Number of fake coins in each bag	0	1	2	3	4 or more
   Observed frequency	43	62	26	13	6
   Expected frequency	44.5	55.7	33.2	12.5	4.1

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
4. Explain why there are 2 degrees of freedom in this case.
5. Given that she obtains a \(\chi ^ { 2 }\) test statistic of 2.67 , test the assistant manager's hypothesis that the binomial distribution is a good model for the number of fake coins in each bag. Use a \(5 \%\) level of significance and state your hypotheses clearly.

5. A research station is doing some work on the germination of a new variety of genetically modified wheat. They planted 120 rows containing 7 seeds in each row.
The number of seeds germinating in each row was recorded. The results are as follows

1. Write down two reasons why a binomial distribution may be a suitable model.
2. Show that the probability of a randomly selected seed from this sample germinating is 0.6 The research station used a binomial distribution with probability 0.6 of a seed germinating. The expected frequencies were calculated to 2 decimal places. The results are as follows

   Number of seeds germinating in each row	0	1	2	3	4	5	6	7
   Expected number of rows	0.20	2.06	\(s\)	23.22	\(t\)	31.35	15.68	3.36

3. Find the value of \(s\) and the value of \(t\).
4. Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the data can be modelled by a binomial distribution.

1. David carries out an experiment with 4 identical dice, each with faces numbered 1 to 6 . He rolls the 4 dice and counts the number of dice showing an even number on the uppermost face. He repeats this 150 times. The results are summarised in the table below.

David defines the random variable \(C\) as the number of dice showing an even number on the uppermost face when the four dice are thrown. David claims that \(C \sim \mathrm {~B} ( 4,0.5 )\)

1. Stating your hypotheses clearly and using a \(1 \%\) level of significance, test David's claim. Show your working clearly. John claims that \(C \sim \mathrm {~B} ( 4 , p )\)
2. Calculate an estimate of the value of \(p\) from the summary of the results of David's experiment. Show your working clearly. John decides to test his claim. He calculates expected frequencies using the results of David's experiment and obtains the following table.

   No. of dice showing an even number	0	1	2	3	4
   Expected frequency	8.65	36.00	\(d\)	39.00	\(e\)

3. Calculate, to 2 decimal places, the value of \(d\) and the value of \(e\)
4. State suitable hypotheses to test John's claim. John obtained a test statistic of 16.9 and carries out a test at the \(1 \%\) level of significance.
5. State what conclusion John should make about his claim.

   END

7. A student collects data on whether competitors in local tennis tournaments are right, or left-handed. The table below shows the number of left-handed players who reached the last 16 for fifty tournaments. The student believes that a binomial distribution with \(n = 16\) and \(p = 0.1\) could be a suitable model for these data.

1. Stating your hypotheses clearly test the student's model at the \(5 \%\) level of significance.
   (13 marks)
   To improve the model the student decides to estimate \(p\) using the data in the table. Using this value of \(p\) to calculate expected frequencies the student had 5 classes after combining and calculated that \(\sum \frac { ( O - E ) ^ { 2 } } { E } = 2.127\)
2. Test at the \(5 \%\) level of significance whether or not the binomial distribution is a suitable model for the number of left-handed players who reach the last 16 in local tennis tournaments. \section*{END}

4 Zara uses a metal detector to search for coins on a beach.
She wonders if the numbers of coins that she finds in an area of \(10 \mathrm {~m} ^ { 2 }\) can be modelled by a Poisson distribution. The table below shows the numbers of coins that she finds in randomly chosen areas of \(10 \mathrm {~m} ^ { 2 }\) over a period of months.

1. Software gives the sample mean as 1.98 and the sample standard deviation as 1.4212. Explain how these values suggest that a Poisson distribution may be an appropriate model for the numbers of coins found. Zara decides to carry out a chi-squared test to investigate whether a Poisson distribution is an appropriate model.
   Fig. 4 is a screenshot showing part of the spreadsheet used to analyse the data. Some values in the spreadsheet have been deliberately omitted. \begin{table}[h]

   	A	B	C	D
   1	Number of coins found	Observed frequency	Expected frequency	Chi-squared contribution
   2	0	13	13.8069	0.0472
   3	1	28
   4	2	30	27.0643	0.3184
   5	3	14	17.8625	0.8352
   6	4	10	8.8419	0.1517
   7	\(\geqslant 5\)	5		0.0015

   \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{table}
2. Showing your calculations, find the missing values in each of the following cells.
   For the rest of this question, you should assume that the number of coins that Zara finds in an area of \(10 \mathrm {~m} ^ { 2 }\) can be modelled by a Poisson distribution with mean 1.98.
   Zara also finds pieces of jewellery independently of the coins she finds. The number of pieces of jewellery that she finds per \(10 \mathrm {~m} ^ { 2 }\) area is modelled by a Poisson distribution with mean 0.42 .
3. Find the probability that Zara finds a total of exactly 3 items (coins and/or jewellery) in an area of \(10 \mathrm {~m} ^ { 2 }\).
4. Find the probability that Zara finds a total of at least 30 items (coins and/or jewellery) in an area of \(100 \mathrm {~m} ^ { 2 }\).

3 Jane wonders whether the number of wasps entering a wasp's nest per 5 second interval can be modelled by a Poisson distribution with mean \(\mu\). She counts the number of wasps entering the nest over 60 randomly selected 5 -second intervals. The results are shown in Fig. 3.1. \begin{table}[h] \end{table}

1. Show that a suitable estimate for the value of \(\mu\) is 5.1. Fig. 3.2 shows part of a screenshot for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]

   	A	B	C	D	E
   \includegraphics[max width=\textwidth, alt={}]{e8624e9b-5143-49d2-9683-cc3a1082694e-4_132_40_1069_273}	Number of wasps	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
   2	\(\leqslant 2\)	7	0.1165	6.9887	0.0000
   3	3	5		8.0874	1.1786
   4	4	12			0.2765
   5	5	10			0.0255
   6	6	10	0.1490	8.9400	0.1257
   7	7	11	0.1086	6.5134	3.0904
   8	\(\geqslant 8\)	5	0.1440	8.6414
   9

   \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{table}
2. Determine the missing values in each of the following cells, giving your answers correct to 4 decimal places.
   Carry out the hypothesis test at the 5\% significance level.
3. Jane also carries out a \(\chi ^ { 2 }\) test for the number of wasps leaving another nest. As part of her calculations, she finds that the probability of no wasps leaving the nest in a 5 -second period is 0.0053 . She finds that a Poisson distribution is also an appropriate model in this case. Find a suitable estimate for the value of the mean number of wasps leaving the nest per 5-second period.

4 Eve lives in a narrow lane in the country. She wonders whether the number of vehicles passing her house per minute can be modelled by a Poisson distribution with mean \(\mu\). She counts the number of vehicles passing her house over 100 randomly selected one-minute intervals. The results are shown in Table 4.1. \begin{table}[h] \end{table}

1. Use the results to find an estimate for \(\mu\). The spreadsheet in Fig. 4.2 shows data for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean from part (a) has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 4.2}

   \multirow[b]{2}{*}{1}	A	B	C	D	E
   	Number of vehicles	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
   2	0	36	0.2725	27.2532	2.8073
   3	1	33	0.3543	35.4291
   4	2	14			3.5400
   5	\(\geqslant 3\)	17			0.5145
   6

   \end{table}
2. Calculate the missing values in each of the following cells, giving your answers correct to 4 decimal places.
   Carry out the \(\chi ^ { 2 }\) test at the 5\% significance level.
3. Eve checks her data and notices that the two largest numbers of vehicles per minute (8 and 10) occurred when some horses were being ridden along the lane, causing delays to the vehicles. She therefore repeats the analysis, missing out these two items of data. She finds that the value of the \(\chi ^ { 2 }\) test statistic is now 4.748. The number of degrees of freedom of the test is unchanged. Make two comments about this revised test.

5. Chris is investigating the distribution of birth months for ice hockey players. He collects data for 869 randomly chosen National Hockey League (NHL) players. He decides to carry out a chi-squared test. Using a spreadsheet, he produces the following output.

1. By considering the output, state the null hypothesis that Chris is testing. State what conclusion Chris should reach and explain why. Chris now wonders if Premier League football players' birth months are distributed uniformly throughout the year. He collects the birth months of 75 randomly selected Premier League footballers. This information is shown in the table below.

   Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
   3	7	11	4	12	2	6	6	5	8	5	6

2. Carry out the chi-squared goodness of fit test at the 10\% significance level that Chris should use to conduct his investigation.

5. John has a game that involves throwing a set of three identical, cubical dice with faces numbered 1 to 6 . He wishes to investigate whether these dice are fair in terms of the number of sixes obtained when they are thrown. John throws the set of three dice 1100 times and records the number of sixes obtained for each throw. The results are shown in the table below. Using these results, conduct a goodness of fit test and draw an appropriate conclusion.

1. A company makes bags. The table below shows the number of bags sold on a random sample of 50 days. A manager believes that the number of bags sold per day can be modelled by the Poisson distribution with mean \(2 \cdot 2\).

1. Carry out a chi-squared goodness of fit test, using a \(10 \%\) significance level.
2. A chi-squared goodness of fit test for the Poisson distribution with mean \(2 \cdot 5\) is conducted. This uses the same number of degrees of freedom as part (a) and gives a test statistic of 1.53 . State, with a reason, which of these two Poisson models is a better fit for the data.

1. In an experiment, James flips a coin 3 times and records the number of heads. He carries out the experiment 100 times with his left hand and 100 times with his right hand.

1. Test, at the \(5 \%\) level of significance, whether or not there is an association between the hand he flips the coin with and the number of heads. You should state your hypotheses, the degrees of freedom and the critical value used for this test.
2. Assuming the coin is unbiased, write down the distribution of the number of heads in 3 flips.
3. Carry out a \(\chi ^ { 2 }\) test, at the \(10 \%\) level of significance, to test whether or not the distribution you wrote down in part (b) is a suitable model for the number of heads obtained in the 200 trials of James' experiment. You should state your hypotheses, the degrees of freedom and the critical value used for this test.

1. Flobee sells tomato seeds in packets, each containing 40 seeds. Flobee advertises that only 4\% of its tomato seeds do not germinate.

Amodita is investigating the germination of Flobee's tomato seeds. She plants 125 packets of Flobee's tomato seeds and records the number of seeds that do not germinate in each packet. Amodita wants to test whether the binomial distribution \(\mathrm { B } ( 40,0.04 )\) is a suitable model for these data. The table below shows the expected frequencies, to 2 decimal places, using this model.

1. Calculate the value of \(r\) and the value of \(s\)
2. Stating your hypotheses clearly, carry out the test at the \(5 \%\) level of significance. You should state the number of degrees of freedom, critical value and conclusion clearly. Amodita believes that Flobee should use a more realistic value for the percentage of their tomato seeds that do not germinate.
   She decides to test the data using a new model \(\mathrm { B } ( 40 , p )\)
3. Showing your working, suggest a more realistic value for \(p\)

1. Table 1 below shows the number of car breakdowns in the Snoreap district in each of 60 months.

\begin{table}[h] \end{table} Anja believes that the number of car breakdowns per month in Snoreap can be modelled by a Poisson distribution. Table 2 below shows the results of some of her calculations. \begin{table}[h] \end{table}

1. State suitable hypotheses for a test to investigate Anja's belief.
2. Explain why Anja has changed the label of the final column to \(\geqslant 5\)
3. Showing your working clearly, complete Table 2
4. Find the value of \(\frac { \left( O _ { i } - E _ { i } \right) ^ { 2 } } { E _ { i } }\) when the number of car breakdowns is
   1. 1
   2. 3
5. Explain why Anja used 3 degrees of freedom for her test. The test statistic for Anja's test is 6.54 to 2 decimal places.
6. Stating the critical value and using a \(5 \%\) level of significance, complete Anja's test.

1. Robin shoots 8 arrows at a target each day for 100 days.

The number of times he hits the target each day is summarised in the table below. Misha believes that these data can be modelled by a binomial distribution.

1. State, in context, two assumptions that are implied by the use of this model.
2. Find an estimate for the proportion of arrows Robin shoots that hit the target. Misha calculates expected frequencies, to 2 decimal places, as follows.

   Number of hits	0	1	2	3	4	5	6	7	8
   Expected frequency	2.81	12.67	\(r\)	28.05	19.73	\(s\)	2.50	0.40	0.03

3. Find the value of \(r\) and the value of \(s\) Misha correctly used a suitable test to assess her belief.
4. 1. Explain why she used a test with 3 degrees of freedom.
   2. Complete the test using a \(5 \%\) level of significance. You should clearly state your hypotheses, test statistic, critical value and conclusion.

1. Liam and Simone are studying the distribution of oak trees in some woodland. They divided the woodland into 80 equal squares and recorded the number of oak trees in each square. The results are summarised in Table 1 below.

\begin{table}[h] \end{table} Liam believes that the oak trees were deliberately planted, with 6 oak trees per square and that a constant proportion \(p\) of the oak trees survived.

1. Suggest the model Liam should use to describe the number of oak trees per square. Liam decides to test whether or not his model is suitable and calculates the expected frequencies given in Table 2. \begin{table}[h]

   Number of oak trees in a square	0 or 1	2	3	4	5	6
   Expected frequency	5.53	14.89	24.26	22.24	10.87	2.21

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
2. Showing your working clearly, complete the test using a \(5 \%\) level of significance. You should state your critical value and conclusion clearly. Simone believes that a Poisson distribution could be used to model the number of oak trees per square. She calculates the expected frequencies given in Table 3. \begin{table}[h]

   Number of oak trees in a square	0 or 1	2	3	4	5	6 or more
   Expected frequency	12.69	16.07	\(s\)	14.58	\(t\)	9.37

   \captionsetup{labelformat=empty} \caption{Table 3}
   \end{table}
3. Find the value of \(s\) and the value of \(t\), giving your answers to 2 decimal places.
4. Write down hypotheses to test the suitability of Simone's model. The test statistic for this test is 8.749
5. Complete the test. Use a \(5 \%\) level of significance and state your critical value and conclusion clearly.
6. Using the results of these tests, explain whether the origin of this woodland is likely to be cultivated or wild.