Information from a sample of 157 restaurant bills collected at the First Crush bistro is available in RestaurantTips. Two intervals are given below for the average tip left at a restaurant; one is a 90% confidence interval and one is a 99% confidence interval Interval A: 3.55 to 4.15 Interval B: 3.35 to 4.35
(a) which one is the 90% confidence interval? | Interv.. which one is the 99% confidence interval?
(b) One waitress generally waits on 20 tables in an average shift. Give a range for her expected daily tip revenue, using both 90% and 99% confidence intervals. Round your answers to the nearest integer.
Confidence level Confidence interval
90% 0.6 to 7.7
99% -1 to 7.7

Confidence Interval for the population mean is basically an interval of range of values where the true population mean can be found with a certain level of confidence.

Mathematically,

Confidence Interval = (Sample mean) ± (Margin of error)

Margin of Error is the width of the confidence interval about the mean.

It is given mathematically as,

Margin of Error = (Critical value) × (standard Error of the mean)

Critical value is obtained from either the t-distribution or the z-distribution tables. It depends on sample size and for smaller sample sizes, whether there is information provided for the population mean and standard deviation.

But whether z-distribution or t-distribution, the critical value increases as the confidence level increases.

Hence, the critical value for the 99% confidence level will be higher than the critical value for the 90% confidence level.

And since all the other parameters that determine the confidence interval and it's width (sample mean and the standard error of the mean) are the same for the 90% and the 99% confidence interval, the larger critical value for the 99% confidence interval means that it has the bigger width.

Hence, of the two intervals given,

Interval A: 3.55 to 4.15

Interval B: 3.35 to 4.35

The larger interval, (3.35, 4.35) is the 99% confidence interval and the smaller interval, (3.55, 4.15) is the 90% confidence interval.

b) The data required for this second part isn't available, but the answer for the confidence interval can be obtained using the steps I have given above and below.

Like I have given above,

Confidence Interval = (Sample mean) ± (Margin of error)

Margin of Error = (Critical value) × (standard Error of the mean)

The sample mean and the standard error of the mean are obtained from the sample data (which is missing).

Sample mean = (Σx)/N

x = each variable

N = number of variables = 20

Standard error of the mean = σₓ = (σ/√n)

σ = standard deviation (obtained from the sample data)

90% critical value for sample size of 20 = 1.73

99% critical value for sample size of 20 = 2.86

Hope this Helps!!!