Find mean from probability statement

Given a normal distribution with known standard deviation and a probability statement involving a specific value, find the unknown mean μ.

7 questions

CAIE S1 2015 June Q1
1 The lengths, in metres, of cars in a city are normally distributed with mean \(\mu\) and standard deviation 0.714 . The probability that a randomly chosen car has a length more than 3.2 metres and less than \(\mu\) metres is 0.475 . Find \(\mu\).
CAIE S1 2015 June Q1
1 The weights, in grams, of onions in a supermarket have a normal distribution with mean \(\mu\) and standard deviation 22. The probability that a randomly chosen onion weighs more than 195 grams is 0.128 . Find the value of \(\mu\).
CAIE S1 2008 November Q3
3
  1. The daily minimum temperature in degrees Celsius ( \({ } ^ { \circ } \mathrm { C }\) ) in January in Ottawa is a random variable with distribution \(\mathrm { N } ( - 15.1,62.0 )\). Find the probability that a randomly chosen day in January in Ottawa has a minimum temperature above \(0 ^ { \circ } \mathrm { C }\).
  2. In another city the daily minimum temperature in \({ } ^ { \circ } \mathrm { C }\) in January is a random variable with distribution \(\mathrm { N } ( \mu , 40.0 )\). In this city the probability that a randomly chosen day in January has a minimum temperature above \(0 ^ { \circ } \mathrm { C }\) is 0.8888 . Find the value of \(\mu\).
OCR S2 2007 January Q1
1 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 22 ) = 0.242\). Find the value of \(\mu\).
Edexcel S1 2011 June Q2
  1. The random variable \(X \sim \mathrm {~N} \left( \mu , 5 ^ { 2 } \right)\) and \(\mathrm { P } ( X < 23 ) = 0.9192\)
    1. Find the value of \(\mu\).
    2. Write down the value of \(\mathrm { P } ( \mu < X < 23 )\).
    3. The discrete random variable \(Y\) has probability distribution
    \(y\)1234
    \(\mathrm { P } ( Y = y )\)\(a\)\(b\)0.3\(c\)
    where \(a , b\) and \(c\) are constants. The cumulative distribution function \(\mathrm { F } ( y )\) of \(Y\) is given in the following table
    \(y\)1234
    \(\mathrm {~F} ( y )\)0.10.5\(d\)1.0
    where \(d\) is a constant.
  2. Find the value of \(a\), the value of \(b\), the value of \(c\) and the value of \(d\).
  3. Find \(\mathrm { P } ( 3 Y + 2 \geqslant 8 )\).
WJEC Further Unit 5 2024 June Q7
7. A farmer uses many identical containers to store four different types of grain: wheat, corn, einkorn and emmer.
  1. The mass \(W\), in kg , of wheat stored in each individual container is normally distributed with mean \(\mu\) and standard deviation \(0 \cdot 6\). Given that, for containers of wheat, \(10 \%\) store less than 19 kg , find the value of \(\mu\).
    The mass \(X\), in kg , of corn stored in each individual container is normally distributed with mean \(20 \cdot 1\) and standard deviation \(1 \cdot 2\).
  2. Find the probability that the mean mass of corn in a random sample of 8 containers of corn will be greater than 20 kg .
    The mass \(Y\), in kg, of einkorn stored in each individual container is normally distributed with mean \(22 \cdot 2\) and standard deviation \(1 \cdot 5\). The farmer and his wife need to move two identical wheelbarrows, one of which is loaded with 3 containers of corn, and the other of which is loaded with 3 containers of einkorn. They agree that the farmer's wife will move the heavier wheelbarrow.
  3. Calculate the probability that the farmer's wife will move
    1. the einkorn,
    2. the corn.
  4. The mass \(E\), in kg , of emmer stored in each individual container is normally distributed with mean \(10 \cdot 5\) and standard deviation \(\sigma\). The farmer's son tries to calculate the probability that the mass of corn in a single container will be more than three times the mass of emmer in a single container. He obtains an answer of 0.35208 .
    1. Find the value of \(\sigma\) that the farmer's son used.
    2. Explain why the value of \(\sigma\) that he used is unreasonable.
      Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}
SPS SPS FM Statistics 2024 September Q4
4. The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 22 ) = 0.242\). Find the value of \(\mu\).
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