Alright, let's carefully go through the problem and its respective parts step-by-step.
To start, we need to understand and organize the given data, summarizing the number of lamps tested and the ranges of their lifetimes:
[tex]\[
\begin{pmatrix}
0 < L \leq 1000 & 30 \\
1000 < L \leq 2000 & 75 \\
2000 < L \leq 3000 & 160 \\
L > 3000 & 35 \\
\end{pmatrix}
\][/tex]
Summarizing the total number of lamps tested:
[tex]\[
\text{Total lamps tested} = 30 + 75 + 160 + 35 = 300
\][/tex]
### Part (a):
Calculate the relative frequency of a lamp lasting for less than 3000 hours, but more than 1000 hours.
1. First, note the number of lamps that lasted between 1000 and 3000 hours:
[tex]\[
\text{Lamps lasting 1000 to 2000 hours} = 75
\][/tex]
[tex]\[
\text{Lamps lasting 2000 to 3000 hours} = 160
\][/tex]
Thus, the total number of lamps lasting between 1000 and 3000 hours is:
[tex]\[
\text{Total lamps lasting between 1000 and 3000 hours} = 75 + 160 = 235
\][/tex]
2. Now, calculate the relative frequency (which is the ratio of the number of occurrences to the total number of events):
[tex]\[
\text{Relative frequency} = \frac{\text{Number of lamps lasting between 1000 and 3000 hours}}{\text{Total number of lamps tested}} = \frac{235}{300}
\][/tex]
By performing the division, we find:
[tex]\[
\text{Relative frequency} \approx 0.7833
\][/tex]
So, the relative frequency of a lamp lasting for less than 3000 hours but more than 1000 hours is approximately 0.7833.
### Part (b):
If a hardware chain ordered 2000 of these lamps, how many would you expect to last for more than 3000 hours?
1. First, find the proportion of lamps that lasted more than 3000 hours:
[tex]\[
\text{Lamps lasting more than 3000 hours} = 35
\][/tex]
[tex]\[
\text{Proportion lasting more than 3000 hours} = \frac{35}{300}
\][/tex]
By performing the division, we get the proportion:
[tex]\[
\text{Proportion lasting more than 3000 hours} \approx 0.1167
\][/tex]
2. Now, use this proportion to determine the expected number of lamps lasting more than 3000 hours out of the 2000 ordered:
[tex]\[
\text{Expected number of lamps} = 2000 \times 0.1167
\][/tex]
By performing the multiplication, we get:
[tex]\[
\text{Expected lamps} \approx 233.33
\][/tex]
So, if a hardware chain ordered 2000 of these lamps, we would expect approximately 233.33 lamps to last more than 3000 hours.
In summary:
- The relative frequency of a lamp lasting between 1000 and 3000 hours is approximately [tex]\(0.7833\)[/tex].
- If a hardware chain ordered 2000 lamps, we would expect approximately 233.33 lamps to last more than 3000 hours.
