Question

**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.

- 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.
**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**±**t**c**s**n*. 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*.

- Explain the meaning of the confidence interval you created.

- 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**±**z**c**σ**n* to
compute the 95% confidence interval. In part b) use =NORM.INV
(.975, 0, 1) to get the z_{c} 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.

Answer #1

1.

__________________________________

Use a t-distribution to find a confidence interval for
the difference in means μd=μ1-μ2 using the relevant sample results
from paired data. Assume the results come from random samples from
populations that are approximately normally distributed, and that
differences are computed using d=x1-x2.
A 99% confidence interval for μd using the paired data in the
following table:
Case
1
2
3
4
5
Treatment 1
22
29
32
25
27
Treatment 2
18
31
26
21
21
Give the best...

Use a t-distribution to find a confidence interval for the
difference in means μd=μ1-μ2 using the relevant sample results from
paired data. Assume the results come from random samples from
populations that are approximately normally distributed, and that
differences are computed using d=x1-x2.
A 95% confidence interval for μd using the paired data in the
following table:
Case
Situation 1
Situation 2
1
77
87
2
81
86
3
95
91
4
61
79
5
71
77
6
72
62...

Problem 1: Confidence Interval for Percent of
Individuals Vaping (there is no data set)
From a July 2019 survey of 1186 randomly selected Americans ages
18-29, it was discovered that 248 of them vaped (used an
e-cigarette) in the past week.
Write one sentence each to check the three conditions of the
Central Limit Theorem. Show your work for the mathematical check
needed to show a large sample size was taken.
Verify your result from part (c) using Stat à...

Use the t-distribution to find a confidence interval
for a difference in means μ1-μ2 given the relevant sample results.
Give the best estimate for μ1-μ2, the margin of error, and the
confidence interval. Assume the results come from random samples
from populations that are approximately normally distributed.
A 90% confidence interval for μ1-μ2 using the sample results
x¯1=81.1, s1=10.3, n1=35 and x¯2=67.1, s2=7.9, n2=20
Enter the exact answer for the best estimate and round your answers
for the margin of...

Use the t-distribution to find a confidence interval
for a mean μ given the relevant sample results. Give the best point
estimate for μ, the margin of error, and the confidence interval.
Assume the results come from a random sample from a population that
is approximately normally distributed.
A 90% confidence interval for μ using the sample results x¯=3.5,
s=0.4, and n=100
Round your answer for the point estimate to one decimal place, and
your answers for the margin of...

Criterion: Calculate a one-sample t
test in SPSS. 6.2
Data: Use the Rex’s Flower Shop data from
problem set 6.1.
Instruction: Complete the following:
Enter the data from problem set 6.1 into SPSS and name the
variable as Roses.
In the Toolbar, click
Analyze, select Compare Means,
and then select One-Sample t
Test.
Select Roses, then click
Arrow to send it over to the right side of the
table. In the box labeled Test Value, enter
8.
Click OK and...

Use the t-distribution to find a confidence interval
for a mean μ given the relevant sample results. Give the best point
estimate for μ, the margin of error, and the confidence interval.
Assume the results come from a random sample from a population that
is approximately normally distributed.
A 95% confidence interval for μ using the sample results x̄ = 79.7,
s = 6.6, and n=42
Round your answer for the point estimate to one decimal place, and
your answers...

Use the t-distribution to find a confidence interval
for a mean μ given the relevant sample results. Give the best point
estimate for μ, the margin of error, and the confidence interval.
Assume the results come from a random sample from a population that
is approximately normally distributed.
A 95% confidence interval for μ using the sample results x̄ = 79.7,
s = 6.6, and n = 42
Round your answer for the point estimate to one decimal place, and...

Use the t-distribution to find a confidence interval
for a mean μ given the relevant sample results. Give the best point
estimate for μ, the margin of error, and the confidence interval.
Assume the results come from a random sample from a population that
is approximately normally distributed.
A 95% confidence interval for μ using the sample results
x̅=10.4, s=5.3, and n=30.
Round your answer for the point estimate to one decimal place,
and your answers for the margin of...

Use the t-distribution to find a confidence interval
for a mean μ given the relevant sample results. Give the best point
estimate for μ, the margin of error, and the confidence interval.
Assume the results come from a random sample from a population that
is approximately normally distributed.
A 90% confidence interval for μ using the sample results x¯=144.2,
s=55.7, and n=50
Round your answer for the point estimate to one decimal place, and
your answers for the margin of...

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