What is this, help me understad ---**Scenario:** A coffee shop owner wants to estimate the average daily sales of their new espresso machine. **Given:** * Sample size: n = 16 days * Sample mean sales: x̄ = $120 * Standard deviation: σ = $30 (known) **Tasks:** 1. Find a 70% confidence interval. 2. Find a 90% confidence interval. 3. Find a 99% confidence interval. 4. Interpret your results.

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Welcome to confidence intervals! Today we'll help a coffee shop owner estimate the average daily sales of their new espresso machine. We have data from 16 days showing an average of 120 dollars with a standard deviation of 30 dollars. Our goal is to find confidence intervals at 70%, 90%, and 99% confidence levels to understand the range of possible true average sales. A confidence interval gives us a range of plausible values for the true population parameter. The formula uses the sample mean plus or minus a margin of error. The margin of error depends on the critical z-value and the standard error. First, we calculate the standard error: sigma divided by the square root of n, which equals 30 dollars divided by 4, giving us 7 dollars and 50 cents. Now we need to find the critical z-values for each confidence level. For 70% confidence, alpha equals 0.30, so we have 0.15 in each tail, giving us z equals 1.04. For 90% confidence, alpha equals 0.10, so we have 0.05 in each tail, giving us z equals 1.645. For 99% confidence, alpha equals 0.01, so we have 0.005 in each tail, giving us z equals 2.576. Notice how higher confidence levels require larger z-values. Now let's calculate the confidence intervals. The margin of error equals z times the standard error. For 70% confidence: margin of error is 1.04 times 7.50 equals 7.80 dollars, giving us the interval from 112.20 to 127.80 dollars. For 90% confidence: margin of error is 1.645 times 7.50 equals 12.34 dollars, giving us 107.66 to 132.34 dollars. For 99% confidence: margin of error is 2.576 times 7.50 equals 19.32 dollars, giving us 100.68 to 139.32 dollars. Notice how wider intervals give us higher confidence. To summarize what we've learned: Confidence intervals provide a range of plausible values for the true population parameter. Higher confidence levels require wider intervals to capture the true parameter. The coffee shop can be 90% confident that true average daily sales are between 107 dollars and 66 cents and 132 dollars and 34 cents. Standard error and critical z-values determine the margin of error for each interval. These intervals help business owners make informed decisions about expected revenue.