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 \(\mu\).

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 .

8 The masses, \(X\) grams, of tomatoes are normally distributed. Half of the tomatoes have masses greater than 56.0 g and \(70 \%\) of the tomatoes have masses greater than 53.0 g .

1. Find the percentage of tomatoes with masses greater than 59.0 g .
2. Find the percentage of tomatoes with masses greater than 65.0 g .
3. Given that \(\mathrm { P } ( a < X < 50 ) = 0.1\), find \(a\).

\(X\) is a continuous random variable such that \(X \sim N(\mu, \sigma^2)\). On the sketch of this Normal distribution in the Printed Answer Booklet, shade the area bounded by the curve, the \(x\)-axis and the lines \(x = \mu \pm \sigma\). [2]