Final answer:
The sample proportion of homes with solar panels in the study is 0.16. A 99% confidence interval for the true proportion can be calculated using the sample proportion, the Z-value for the 99% confidence level, and the sample size. This interval will estimate the proportion of all homes in the city with solar panels with 99% confidence.
Step-by-step explanation:
The question you have asked pertains to constructing a 99% confidence interval for the proportion of all homes in a city that have solar panels installed on their roofs, based on a sample. The sample proportion is the first step in this calculation, which is the number of homes with solar panels divided by the total number of homes sampled. In your case, the sample proportion (p-hat) is 24 divided by 150, which is 0.16.
To construct a confidence interval, we need to use the formula:
p-hat ± Z*(sqrt((p-hat*(1-p-hat))/n))
Where:
p-hat is the sample proportion, which is 0.16.Z is the Z-value corresponding to the 99% confidence level, which is typically 2.576.n is the sample size, which is 150.
After plugging these values into the formula, we calculate the margin of error and then add and subtract it from the sample proportion to get the lower and upper bounds of the confidence interval. This interval gives us a range in which we are 99% confident that the true proportion of homes with solar panels in the city lies.