Questions: A client of a commercial rose grower has been keeping records on the shelf-life of a rose. The client sent the following frequency distribution to the grower. Rose Shelf-Life Days of Shelf-Life Frequency(f) 1 - 6 5 7 - 12 3 13 - 18 7 19 - 24 8 25 - 30 2 31 - 36 2 Step 1 of 2 : Calculate the population mean for the shelf-life. Round your answer to two decimal places, if necessary.

Solution Steps

To find the population mean of the rose shelf-life, we first calculate the midpoints for each class interval:

\[ \begin{align_} \text{Midpoint of } 1 - 6 & : \frac{1 + 6}{2} = 3.5 \\ \text{Midpoint of } 7 - 12 & : \frac{7 + 12}{2} = 9.5 \\ \text{Midpoint of } 13 - 18 & : \frac{13 + 18}{2} = 15.5 \\ \text{Midpoint of } 19 - 24 & : \frac{19 + 24}{2} = 21.5 \\ \text{Midpoint of } 25 - 30 & : \frac{25 + 30}{2} = 27.5 \\ \text{Midpoint of } 31 - 36 & : \frac{31 + 36}{2} = 33.5 \\ \end{align_} \]

Next, we calculate the sum of the products of midpoints and their corresponding frequencies:

\[ \begin{align_} \text{Sum} & = (3.5 \times 5) + (9.5 \times 3) + (15.5 \times 7) + (21.5 \times 8) + (27.5 \times 2) + (33.5 \times 2) \\ & = 17.5 + 28.5 + 108.5 + 172 + 55 + 67 \\ & = 449.5 \\ \end{align_} \]

We then calculate the total number of observations (frequencies):

\[ N = 5 + 3 + 7 + 8 + 2 + 2 = 27 \]

Now, we can calculate the population mean using the formula:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{449.5}{27} \approx 16.61 \]

Final Answer