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# Problem #1 Confidence Interval for Means using the t and z Distribution.    Psychologists studied the percent...

Problem #1 Confidence Interval for Means using the t and z Distribution.    Psychologists studied the percent tip at a restaurant when a message indicating that the next day’s weather would be nice was written on the bill. Here are tips from a random sample of patrons who received such a bill, measured in percent of the total bill:

20.8     18.7     19.9     20.6     21.9     23.4     22.8     24.9     22.2     20.3

24.9     22.3     27.0     20.4     22.2     24.0     21.1     22.1     22.0     22.7

Open an Excel file, go to File—Options—Add-ins. At the bottom of the box that opens, next to Manage: Excel Add-ins, click GO. Check the box labelled Analysis ToolPak and click OK. This gives you access to a new set of Excel functions under the Data tab on the far right titled Data Analysis.

1. Enter the data set above in one column in Excel. Go to the Data tab, then to Data Analysis. Highlight Descriptive Statistics and click OK. For the input range, highlight your data set. Check the boxes marked summary statistics and confidence level for the mean, then click OK. Excel will open a new sheet that contains a bunch of descriptive statistics from the data set (some of which you’ll recognize and some you won’t). You’ll want to widen columns A and B to better see the information. The tabs at the bottom of the Excel Workbook allow you to toggle back and forth between sheets.
2. Using t distribution. Construct a 95% confidence interval for the mean percent tip at this restaurant when a message indicating that the next day’s weather will be nice is written on the bill. Find the sample mean, sample standard deviation and sample size in the output from part a. Since we do not know the population standard deviation, the critical value for the confidence interval will be a value of t and you need to use the formula x±tcsn . Find the appropriate value of t using the formula t.inv.2T(.05,19). Compare with the value from the t table. Compute the margin of error in Excel—you can reference the cells on the other sheet to do so. Round the margin of error to six decimal places and write your final answer below in the form x±margin of error .
1. Explain the meaning of the confidence interval you created.
1. A second way to use Excel to create the same confidence interval you made in part a. First, enter the data in one column in Excel. Click on an empty cell and hit the fx button. Under the statistical category, highlight CONFIDENCE.T and click OK. For the box labeled alpha, enter .05 because for 95% confidence, alpha is .05 (this is like a two-tailed hypothesis test when α=.05 ). For the box labeled standard deviation, enter the sample standard deviation found in part a). In the box labeled size, enter the sample size. Click OK. Excel returns the margin of error, not an actual confidence interval. Write down your confidence interval below in the form x±margin of error . Use excel to compute the lower and upper limit of the confidence interval.    Round the margin of error to six decimal places.

e) Using Normal Distribution (z-score). Repeat steps b) and d) with the following changes.

Now use the formula x±zcσn to compute the 95% confidence interval. In part b) use =NORM.INV (.975, 0, 1) to get the zc value and in part c) use CONFIDENCE.NORM (.05, σ, sample size) to compute the margin of error. Is there a difference in the confidence intervals using the t-score and the z-score?

Again compute x±E to get the confidence interval.

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