SPS SPS FM Statistics (SPS FM Statistics) 2022 February

1. At a seaside resort the number \(X\) of ice-creams sold and the temperature \(Y ^ { \circ } \mathrm { F }\) were recorded on 20 randomly chosen summer days. The data can be summarised as follows.

$$\sum x = 1506 \quad \sum x ^ { 2 } = 127542 \quad \sum y = 1431 \quad \sum y ^ { 2 } = 104451 \quad \sum x y = 111297$$

1. Calculate the equation of the least squares regression line of \(y\) on \(x\), giving your answer in the form \(y = a + b x\).
2. Explain the significance for the regression line of the quantity \(\sum \left[ y _ { i } - \left( a x _ { i } + b \right) \right] ^ { 2 }\).
3. It is decided to measure the temperature in degrees Centigrade instead of degrees Fahrenheit. If the same temperature is measured both as \(f ^ { \circ }\) Fahrenheit and \(c ^ { \circ }\) Centigrade, the relationship between \(f\) and \(c\) is \(c = \frac { 5 } { 9 } ( f - 32 )\). Find the equation of the new regression line.
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2. When babies are born, their head circumferences are measured. A random sample of 50 newborn female babies is selected. The sample mean head circumference is 34.711 cm . The sample standard deviation head circumference is 1.530 cm .

1. Determine a \(95 \%\) confidence interval for the population mean head circumference of newborn female babies.
2. Explain why you can calculate this interval even though the distribution of the population of head circumferences of newborn female babies is unknown.
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3. In air traffic management, air traffic controllers send radio messages to pilots. On receiving a message, the pilot repeats it back to the controller to check that it has been understood correctly. At a particular site, on average \(4 \%\) of messages sent by controllers are not repeated back correctly and so have been misunderstood. You should assume that instances of messages being misunderstood occur randomly and independently.

1. Find the probability that exactly 2 messages are misunderstood in a sequence of 50 messages.
2. Find the probability that in a sequence of messages, the 10th message is the first one which is misunderstood.
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3. Find the probability that in a sequence of 20 messages, there are no misunderstood messages. [1]
4. Determine the expected number of messages required for 3 of them to be misunderstood.
5. Determine the probability that in a sequence of messages, the 3rd misunderstood message is the 60th message in the sequence.
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4. Members of a photographic group may enter a maximum of 5 photographs into a members only competition.
Past experience has shown that the number of photographs, \(N\), entered by a member follows the probability distribution shown below. Given that \(\mathrm { E } ( 4 N + 2 ) = 14.8\) and \(\mathrm { P } ( N = 5 \mid N > 2 ) = \frac { 1 } { 2 }\)

1. show that \(\operatorname { Var } ( N ) = 2.76\) The group decided to charge a 50 p entry fee for the first photograph entered and then 20 p for each extra photograph entered into the competition up to a maximum of \(\pounds 1\) per person. Thus a member who enters 3 photographs pays 90 p and a member who enters 4 or 5 photographs just pays £l Assuming that the probability distribution for the number of photographs entered by a member is unchanged,
2. calculate the expected entry fee per member.
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5. A practice examination paper is taken by 500 candidates, and the organiser wishes to know what continuous distribution could be used to model the actual time, \(X\) minutes, taken by candidates to complete the paper. The organiser starts by carrying out a goodness-of-fit test for the distribution \(\mathrm { N } \left( 100,15 ^ { 2 } \right)\) at the \(5 \%\) significance level. The grouped data and the results of some of the calculations are shown in the following table.

1. State suitable hypotheses for the test.
2. Show how the figures 123.754 and 0.222 in the column for \(100 \leqslant X < 110\) were obtained. [3]
3. Carry out the test.
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6. The continuous random variable \(Y\) has a uniform distribution on [ 0,2 ].

1. It is given that \(\mathrm { E } [ a \cos ( a Y ) ] = 0.3\), where \(a\) is a constant between 0 and 1 , and \(a Y\) is measured in radians. Determine the value of the constant \(a\).
2. Determine the \(60 ^ { \text {th } }\) percentile of \(Y ^ { 2 }\).
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