Symmetric properties of normal

6 A farmer finds that the weights of sheep on his farm have a normal distribution with mean 66.4 kg and standard deviation 5.6 kg .

1. 250 sheep are chosen at random. Estimate the number of sheep which have a weight of between 70 kg and 72.5 kg .
2. The proportion of sheep weighing less than 59.2 kg is equal to the proportion weighing more than \(y \mathrm {~kg}\). Find the value of \(y\). Another farmer finds that the weights of sheep on his farm have a normal distribution with mean \(\mu \mathrm { kg }\) and standard deviation 4.92 kg . 25\% of these sheep weigh more than 67.5 kg .
3. Find the value of \(\mu\).

3 The random variable \(G\) has a normal distribution. It is known that $$\mathrm { P } ( G < 56.2 ) = \mathrm { P } ( G > 63.8 ) = 0.1 \text {. }$$ Find \(\mathrm { P } ( G > 65 )\).

1. The random variable \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\)

Given that \(\mathrm { P } ( X > \mu + a ) = 0.35\) where \(a\) is a constant, find

1. \(\mathrm { P } ( X > \mu - a )\)
2. \(\mathrm { P } ( \mu - a < X < \mu + a )\)
3. \(\mathrm { P } ( X < \mu + a \mid X > \mu - a )\)

7. The random variable \(X\) is normally distributed with mean 79 and variance 144 . Find

1. \(\mathrm { P } ( X < 70 )\),
2. \(\mathrm { P } ( 64 < X < 96 )\). It is known that \(\mathrm { P } ( 79 - a \leq X \leq 79 + b ) = 0.6463\). This information is shown in the figure below.
   \includegraphics[max width=\textwidth, alt={}, center]{df898ff4-c3ef-400c-b4f7-f4df3757941d-6_581_983_818_590} Given that \(\mathrm { P } ( X \geq 79 + b ) = 2 \mathrm { P } ( X \leq 79 - a )\),
3. show that the area of the shaded region is 0.1179 .
4. Find the value of \(b\).

3. The random variable \(Y\) has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) The \(\mathrm { P } ( Y > 17 ) = 0.4\) Find

1. \(\mathrm { P } ( \mu < Y < 17 )\)
2. \(\mathrm { P } ( \mu - \sigma < Y < 17 )\)

15 You must show detailed reasoning in this question. The screenshot in Fig. 15 shows the probability distribution for the continuous random variable \(X\), where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). \begin{figure}[h] \end{figure} The distribution is symmetrical about the line \(x = 35\) and there is a point of inflection at \(x = 31\).
Fifty independent readings of \(X\) are made. Show that the probability that at least 45 of these readings are between 30 and 40 is less than 0.05 .

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1. 
   \end{tabular} &

   The mass of aluminium cans recycled each day in a city may be modelled by a normal distribution with mean 24500 kg and standard deviation 5200 kg .

   State the probability that the mass of aluminium cans recycled on any given day is not equal to 24500 kg .

   [1 mark]

   
   \hline \end{tabular} \end{center} 14
2. A member of the council claims that if a different sample of 24 days had been used the hypothesis test in part (b) would have given the same result. Comment on the validity of this claim.

17 In 2019, the lengths of new-born babies at a clinic can be modelled by a normal distribution with mean 50 cm and standard deviation 4 cm . 17

1. This normal distribution is represented in the diagram below. Label the values 50 and 54 on the horizontal axis. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{deec0d32-b031-4227-bc80-7150a0acbc94-29_375_531_644_817} \captionsetup{labelformat=empty} \caption{Length (cm)}
   \end{figure} 17
2. State the probability that the length of a new-born baby is less than 50 cm .
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3. Find the probability that the length of a new-born baby is more than 56 cm .
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4. Find the probability that the length of a new-born baby is more than 40 cm but less than 60 cm .
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5. Determine the length exceeded by 95\% of all new-born babies at the clinic.
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6. In 2020, the lengths of 40 new-born babies at the clinic were selected at random.
   The total length of the 40 new-born babies was 2060 cm .
   Carry out a hypothesis test at the \(10 \%\) significance level to investigate whether the mean length of a new-born baby at the clinic in 2020 has increased compared to 2019. You may assume that the length of a new-born baby is still normally distributed with standard deviation 4 cm .