1 Over time LED light bulbs gradually lose brightness. For a particular type of LED bulb, it is known that the mean reduction in brightness after 10000 hours is \(2.6 \%\). A manufacturer produces a new version of this bulb, which costs less to make, but is claimed to have the same reduction in brightness after 10000 hours as the previous version.
In order to check this claim, a random sample of 10 bulbs is selected. For each bulb, the original brightness and the brightness after 10000 hours are measured, in suitable units. A spreadsheet is used to produce a \(95 \%\) confidence interval for the mean percentage reduction in brightness. A screenshot of the spreadsheet is shown in Fig. 1.
\begin{table}[h]
| A | B | C | D | E | F | G | H | 1 | J | K |
| 1 | Original brightness | 1075 | 1121 | 1106 | 1095 | 1101 | 1109 | 1114 | 1123 | 1108 | 1115 |
| 2 | After 10000 hours | 1042 | 1084 | 1076 | 1065 | 1070 | 1079 | 1081 | 1091 | 1080 | 1082 |
| 3 | Percentage reduction | 3.07 | 3.30 | 2.71 | 2.74 | 2.82 | 2.71 | 2.96 | 2.85 | 2.53 | 2.96 |
| 4 | | | | | | | | | | | |
| 5 | | | | | | | | | | | |
| 6 | Sample mean (\%) | 2.8650 | | | | | | | | | |
| 7 | Sample sd (\%) | 0.2179 | | | | | | | | | |
| 8 | SE | 0.0689 | | | | | | | | | |
| 9 | DF | 9 | | | | | | | | | |
| 10 | tvalue | 2.262 | | | | | | | | | |
| 11 | Lower limit | 2.709 | | | | | | | | | |
| 12 | Upper limit | 3.021 | | | | | | | | | |
| 1.3 | | | | | | | | | | | |
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{table}
- State the confidence interval in the form \(a < \mu < b\).
- Explain whether the confidence interval suggests that the mean percentage reduction in brightness after 10000 hours is different from 2.6\%.
- Explain how the value in cell B8 was calculated.
- State an assumption necessary for this confidence interval to be calculated.
- Explain the advantage of using the same bulbs for both measurements.