Researchers investigated whether providing a fancy, foil-wrapped piece of chocolate with the dinner bill would lead to higher tips than not providing such a treat. Sixty-four dinner parties at a restaurant in Ithaca, NY were randomly assigned to receive such a piece of chocolate or not with their dinner bill. Of the 32 parties who received the chocolate, the average tip (as a percentage of the bill) was 17.84%, with a standard deviation of 3.06%. Of the 32 parties who did not receive the chocolate, the average tip (as a percentage of the bill) was 15.06%, with a standard deviation of 1.89%. Determine a 96% confidence interval for the size of the difference in average tips between the two groups

Answer:

The 96% confidence interval for the size of the difference in average tips between the two groups is (1.47%, 4.09%).

Step-by-step explanation:

Before building the confidence interval, we need to understand the central limit theorem and subtraction of normal variables.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Subtraction of normal variables:

When we subtract normal variables, the mean is the subtraction of the means while the standard deciation is the square root of the sum of the variances.

Of the 32 parties who received the chocolate, the average tip (as a percentage of the bill) was 17.84%, with a standard deviation of 3.06%.

This means that:

[tex]\mu_{C} = 17.84, s_{C} = \frac{3.06}{\sqrt{32}} = 0.541[/tex]

Of the 32 parties who did not receive the chocolate, the average tip (as a percentage of the bill) was 15.06%, with a standard deviation of 1.89%.

This means that:

[tex]\mu_{NC} = 15.06, s_{NC} = \frac{1.89}{\sqrt{32}} = 0.3341[/tex]

Difference in average tips

The distribution has mean:

[tex]\mu = \mu_{C} - \mu_{NC} = 17.84 - 15.06 = 2.78[/tex]

Standard deviation:

[tex]s = \sqrt{s_{C}^2 + s_{NC}^2} = \sqrt{0.541^2 + 0.3341^2} = 0.6358[/tex]

96% confidence interval

We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:

[tex]\alpha = \frac{1 - 0.96}{2} = 0.02[/tex]

Now, we have to find z in the Ztable as such z has a pvalue of [tex]1 - \alpha[/tex].

That is z with a pvalue of [tex]1 - 0.02 = 0.98[/tex], so Z = 2.054.

Now, find the margin of error M as such

[tex]M = zs[/tex]

[tex]M = 2.054*0.6358 = 1.31[/tex]

The lower end of the interval is the sample mean subtracted by M. So it is 2.78% - 1.31% = 1.47%

The upper end of the interval is the sample mean added to M. So it is 2.78% + 1.31% = 4.09%

The 96% confidence interval for the size of the difference in average tips between the two groups is (1.47%, 4.09%).

what is the probability of rolling an even number with one roll of a number cube?
​

Criteria are the goals that a design must achieve. They are determined by what you want the cup to do. Criteria for the volume of the cup compartments have been decided for you. The volume is expressed in four different units: Volume cubic centimeters milliliters fluid ounces cups cold pack 150 150 5 0.6 drink compartment 600 600 20 2.5 total 750 750 25 3.1 Although the volume is set, the shape of the cup is up to you. There are no right or wrong answers. Do you want your cup to be tall and skinny or short and wide

do this for me? helppp all my point

My cousin made 50 muffins and brought 3/5 of them to work how many did she leave at home?

The sales tax in your city is 8.8% and an item cost $63 before tax.

How much tax would you pay on that item?
Round to the nearest hundredth or cent.

There are 20 children at a library. Six of the children have a averahe age of 5 years. The remaining children have a sverage age of 8 years. What is the average age of the children at the library?​

A football stadium contain 80% of its full capacity of spectators for a grand final match. At the end of the game, spectators leave the ground through various exits at the rate of 5 spectators per seconds. After 20 minutes the stadium contain 60% of its full capacity of spectators.
A. Calculate the full capacity of the stadium
B. How many spectators will be in the stadium 1 hour after the end of the game?

find the volume of a rectangular prism with a base area of 13 ft.² and a height Of 7 feet

Consider the following diagram below.
D
(4x-3)
(39)
(5y)
(6x+13)
F
Find the values of cand y.

We are told that the data is representative of the two populations (U.S. males aged 20-29 years and U.S. males aged 75+ years), and we will assume that researchers collected random samples. The samples are very large; therefore, the conditions are met for use of the T-test. Using StatCrunch, we find a T-score of 5.3 and a P-value of "< 0.0001." What can we conclude? Group of answer choices The data prove that, in the U.S., young men weigh about 4.9 kgs more on average than old men. The data suggest that young men weigh more on average than old men in the U.S., but that the difference in mean weights for the two groups is not statistically significant. The data provide strong evidence that young men weigh more on average than old men in the U.S. no conclusion is possible because we did not specify a significance level

k over 1.5 equals 3 what is k

write the sentence as an inequality. A number x is greater than 6 but less than 10. ​

If cosx=-3/7, find the value for cos4x

According to one study, the average monthly cell phone bill in the U.S. is $70. If a 20-year old student with an average bill gives up her cell phone and each month invests the $70 she would have spent on her phone bill in a savings plan that averages a 4% annual return, how much will she have saved by the time she is 65?

Dulce the dog loves to go on walks. Each morning she walks for 45minutes.She also goes for a 20 minute walk at lunch, and an evening stroll for 1 hour and 30 minutes. How long does she walk each day?

11x + x helppp plzzz

Antonio buys 2.25 pounds of grapes at the store. The grapes are on sale for $1.20 per pound. How much will Antonio pay for the grapes?

What is the length from Alaska to Chicago times 5 divided by 3?

In the Special Right Triangle below, solve for the value of y. Y = _____ (keep your answer in simplest radical form)

Options
6
6√3
6√6
12
12√2
18
12√3

solve for x in kite ​

The cost of dell personal computer is $40less than the cost of a gateway computer, while and epson computer cost $70 more than a gateway. If an order of three gateways, one dell and three espoused total $6820, what is the cost of each type of computer?

Which data set has a median of 15?
18, 9, 19, 16, 6, 20. 13
9, 17, 13, 15, 16, 8, 12
18, 15, 11, 14, 19, 15, 6
7,16, 14, 16, 11, 7, 17

PLEASE HELP!!!!!!!!!!

The Expression
12
3+9
is what:
A: Sum
B. quotient
C. difference

Lola has 3 feet of ribbon. She needs to cut the ribbon into equal 2-inch strips
2
How many 2-inch strips can she cut from the 3 feet of ribbon?
2
A
15
B
90
C
21
D
145

4. Ms. Hwang started reading at 3:35 p.m. She finished reading at 4:16 p.m.
How many minutes did Ms. Hwang read?

The price of milk has been increasing by 4% per year. If he gallon of milk cost exactly 3 dollars right now, how much should it cost in 4 years?

Which number is irrational

PLEASE HELP! I WILL GIVE YOU BRAINLIEST

If F(x) = x - 5 and G(x) = x2, what is G(F(x))?

Select the expression that represents the following statement: 3 times one fourth the difference of 26 and 10. (4 points)

one fourth x (26 + 10) x 3
one fourth x (26 − 10) x 3
3 x one fourth x 26 − 10
3 x one fourth x 26 x 10