Final answer:
A 90% confidence interval for the quartet's income stream from wedding events can be calculated using the mean and standard deviation of their sample data, along with the t-score for the appropriate confidence level and degrees of freedom.
Step-by-step explanation:
To construct a 90% confidence interval for the quartet's income stream based on the number of weddings played over six months, first, we need to find the mean (μ) and standard deviation (s) of the sample data: 3, 5, 7, 4, 4, and 5.
The mean (μ) of the sample is calculated as (3 + 5 + 7 + 4 + 4 + 5)/6 = 28/6 = 4.67. The sample standard deviation (s) is calculated using the formula for s = √[Σ(xi - μ)2/(n - 1)], where xi are the samples, μ is the mean, and n is the number of samples. Plugging in our values gives us a standard deviation (s).
Next, we use the standard error of the mean (SE), which is s/√(n), and find the t-score that corresponds to our 90% confidence level for a t-distribution with n-1 degrees of freedom (dof). Using a t-table or an online calculator, we find the t-score.
Finally, our confidence interval (CI) can be expressed as:
CI = μ ± (t * SE)
Plug in the values for μ, t, and SE to calculate the 90% CI. For example, if our calculated standard deviation (s) is 1.5 and our t-score is 2.015, the margin of error (ME) would be 2.015*(1.5/√(6)) and the CI would be 4.67 ± ME.
Thus, the confidence interval provides a range in which the quartet can expect the true mean of weddings they play at to fall, with 90% certainty.