Question

people tend to be more generous after receiving good news. are they less generous after receiving bad news? the average tip left by adult americans is 20%. say that a server gives 20 patrons of a restaurant a message on their bill warning them that tomorrow’s weather will be bad and record the percentage tip they leave. the tips as a percentage of the total bill are given in the table. tips (percentages) 18.0 14.0 19.1 17.0 19.2 13.6 18.8 17.5 18.4 20.0 19.0 20.2 18.5 16.1 18.0 16.8 21.2 18.2 19.4 suppose that percentage tips are normal with \\( \sigma = 2 \\). click to download the data in your preferred format. csv excel jmp mac-text minitab14-18 minitab18+ pc-text r spss ti crunchit! information about the naep test can be found online at https://www.nationsreportcard.gov. solve: we have a random sample of \\( n = 20 \\) customers and were told to assume tips have a normal distribution. find \\( \bar{x} \\). give your answer to two decimal places. \\( \bar{x} = \\) % find the z statistic. give your answer to two decimal places. \\( z = \\)

Explanation:

Step1: Calculate the sample mean ($\bar{x}$)

First, we list out all the tip percentage values from the table: 18.0, 19.1, 19.2, 18.3, 18.4, 19.0, 18.5, 16.1, 16.8, 18.2, 14.0, 17.0, 13.6, 17.5, 20.0, 20.2, 18.8, 18.0, 21.2, 19.4.

To find the mean, we sum all these values and divide by the number of observations ($n = 20$).

Sum of the values:

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Now, the sample mean $\bar{x} = \frac{\text{Sum}}{n} = \frac{361.3}{20} = 18.065 \approx 18.07$ (rounded to two decimal places).

Step2: Calculate the z - statistic

The formula for the z - statistic (assuming a one - sample z - test, since we know the population standard deviation $\sigma = 2$) is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$, where $\mu$ is the population mean (here, we are testing against the idea of the average tip left by adult Americans, which is 20%? Wait, no, wait. Wait, the problem says "the average tip left by adult Americans is 20%". Wait, let's re - check. Wait, the problem: "People tend to be more generous after receiving good news. Are they less generous after receiving bad news? The average tip left by adult Americans is 20%. Say that a server gives 20 patrons of a restaurant a message on their bill warning them that tomorrow’s weather will be bad and record the percentage tip they leave."

So, $\mu = 20$, $\sigma = 2$, $n = 20$, $\bar{x}=18.07$

So, $z=\frac{18.07 - 20}{\frac{2}{\sqrt{20}}}$

First, calculate the denominator: $\frac{2}{\sqrt{20}}\approx\frac{2}{4.4721}\approx0.4472$

Then, the numerator: $18.07 - 20=- 1.93$

Then, $z=\frac{-1.93}{0.4472}\approx - 4.32$ (rounded to two decimal places)

Answer:

First, the sample mean $\bar{x}\approx18.07$ (if the first part is to find $\bar{x}$) and the z - statistic $z\approx - 4.32$ (if the second part is to find $z$). If we assume the first blank is for $\bar{x}$: $\bar{x}=\boxed{18.07}$ and for $z$: $z=\boxed{-4.32}$ (depending on which part the question is asking for. If it's two parts, first the sample mean, then the z - statistic).