7. A lightbulb manufacturer wants to make sure the lightbulbs being produced give o the correct level of
brightness. Brightness is measured using the unit of measurement known as a lumen. A worker
randomly selects 100 lightbulbs and tests how bright they are. The mean brightness of this sample of
lightbulbs is 450 lumens with a standard deviation of 6.8 lumens. The worker uses this data to construct
the following 95% confidence interval: (448.67,451.33)
a. Interpret this confidence interval in context of the problem.
This confidence interval tells us that we can be 95% confident that the true mean brightness of all the lightbulbs produced by the manufacturer is between 448.67 and 451.33 lumens, based on the sample of 100 lightbulbs tested by the worker.
b. What is the margin of error for this confidence interval?
The margin of error for this confidence interval is (451.33 - 448.67)/2 = 1.33/2 = 0.665 lumens.
c. If the lightbulb manufacturer wants to increase the precision of their estimates, what can they do?
To increase the precision of their estimates, the lightbulb manufacturer can either increase the sample size of the lightbulbs tested or decrease the standard deviation of the brightness measurements. A larger sample size or a smaller standard deviation will result in a narrower confidence interval and therefore a more precise estimate of the true mean brightness of all the lightbulbs produced.
To construct a 95% confidence interval for the mean brightness of the lightbulbs, the worker used the sample mean and standard deviation. Here are the steps to calculate the confidence interval:
1. Determine the sample size (n):
In this case, the sample size is 100 lightbulbs.
2. Calculate the sample mean (x̄):
The sample mean brightness is given as 450 lumens.
3. Calculate the standard deviation (s):
The standard deviation of the sample brightness is given as 6.8 lumens.
4. Determine the desired level of confidence (C):
In this case, the desired level of confidence is 95%, which corresponds to a significance level (α) of 0.05.
5. Find the critical value (z):
The critical value is determined based on the confidence level and the distribution. For a 95% confidence level, the critical value is typically 1.96 for large sample sizes (n ≥ 30).
6. Calculate the margin of error (E):
The margin of error is given by the formula:
E = z * (s / √n)
Using the given values, we have:
E = 1.96 * (6.8 / √100) ≈ 1.33
7. Calculate the confidence interval:
The confidence interval is calculated as:
(x̄ -E, x̄ +E),
where x̄ is the sample mean and E is the margin of error.
Using the given values, the confidence interval is:
(450 - 1.33, 450 + 1.33) ≈ (448.67, 451.33)
Therefore, the worker constructed a 95% confidence interval for the mean brightness of the lightbulbs as (448.67, 451.33) lumens.
To understand how the worker constructed the 95% confidence interval for the brightness of the lightbulbs, we need to know a few concepts: sample mean, standard deviation, confidence interval, and the appropriate formula.
First, let's define the terms:
- Sample mean: It is the average value of a sample. In this case, it is the mean brightness of the 100 lightbulbs, which is 450 lumens.
- Standard deviation: It measures the variability or spread of the data. In this case, it is the standard deviation of the sample brightness of the 100 lightbulbs, which is 6.8 lumens.
- Confidence interval: It is a range of values within which we expect the population parameter to lie. In this case, it is the interval within which we expect the true mean brightness of all the lightbulbs to lie.
Now, let's explain how the worker constructed the confidence interval:
1. The worker used a formula to calculate the margin of error, which is a measure of the uncertainty in estimating the population mean. The formula is:
Margin of Error = Critical Value x Standard Deviation / sqrt(sample size)
Here, the critical value corresponds to the level of confidence. For a 95% confidence interval, the critical value is approximately 1.96 (which can be found in statistical tables or calculated using software).
2. To find the margin of error, we substitute the values into the formula:
Margin of Error = 1.96 x 6.8 / sqrt(100)
3. The worker then calculated the lower limit and upper limit of the confidence interval using the mean brightness and the margin of error:
Lower Limit = Sample Mean - Margin of Error
Lower Limit = 450 - (1.96 x 6.8 / sqrt(100))
Upper Limit = Sample Mean + Margin of Error
Upper Limit = 450 + (1.96 x 6.8 / sqrt(100))
After performing the calculations, the worker obtained a 95% confidence interval of (448.67, 451.33) lumens. This means that based on the sample of 100 lightbulbs, we can be 95% confident that the true mean brightness of all lightbulbs falls within this interval.