Answer

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**Hint:**To prepare the frequency chart of the above data, we have to count the number of times a particular temperature appears. The maximum and the minimum temperature is the highest and the lowest value of the temperature recorded. And the range of temperature is writing the lowest and highest value together with (hyphen) separating them. Now, the mean temperature of the month is calculated by using the following formula: $Mean=\dfrac{\text{Sum of observations}}{\text{Total number of observations}}$.

**Complete step by step answer:**

The table of the temperatures given in the above problem is as follows:

31 | 35 | 32 | 31 | 38 | 30 | 36 | 35 | 32 | 35 | 38 | 33 | 33 | 34 | 38 |

32 | 30 | 31 | 32 | 36 | 37 | 34 | 31 | 34 | 35 | 30 | 32 | 31 | 34 | 35 |

Solving part (a) of the above problem we get,

We are going to make a table of the temperature and the frequency of that particular temperature. Now, frequency is the number of times a particular observation occurs.

Temperature$\left( ^{\circ }C \right)$ | Frequency |

30 | 3 |

31 | 5 |

32 | 5 |

33 | 2 |

34 | 4 |

35 | 5 |

36 | 2 |

37 | 1 |

38 | 3 |

Hence, we have solved part (a).

Solving the part (b) we get,

The maximum temperature recorded in the month of September is ${{38}^{\circ }}C$.

Solving the part (c) we get,

The minimum temperature recorded in the month of September is ${{30}^{\circ }}C$.

Solving the part (d) we get,

The range of the temperature recorded in the month of September is equal to: $\left( {{30}^{\circ }}C-{{38}^{\circ }}C \right)$.

Solving the part (e) we get,

We are asked to find the mean temperature recorded in the month of September which we are going to find by using the following formula of the mean:

$Mean=\dfrac{\text{Sum of observations}}{\text{Total number of observations}}$

Total number of observations is calculated by adding the frequencies which we have written above. Adding the frequencies of the temperatures we get,

Frequency |

3 |

5 |

5 |

2 |

4 |

5 |

2 |

1 |

3 |

Total:30 |

From the above, we got the total number of the observations as 30.

Now, we are going to find the sum of all the observations by adding all the observations:

We are going to add the above observations by multiplying frequency of the given temperature from the table which we have drawn above:

$\begin{align}

& 30\left( 3 \right)+31\left( 5 \right)+32\left( 5 \right)+33\left( 2 \right)+34\left( 4 \right)+35\left( 5 \right)+36\left( 2 \right)+37\left( 1 \right)+38\left( 3 \right) \\

& =90+155+160+66+136+175+72+37+114 \\

& =1005 \\

\end{align}$

Now, we are going to substitute the value of sum of all observations and the total number of observations in the formula for mean and we get,

$\begin{align}

& Mean=\dfrac{1005}{30} \\

& \Rightarrow Mean={{33.5}^{\circ }}C \\

\end{align}$

Hence, we have found the mean of the temperatures recorded in the month of September.

**Note:**Instead of the way we have added the observations in the above solution, we can just add all the observations one by one. We did the addition which we have shown above in that manner because it will reduce the time needed to do the addition.

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