Question 1201519: Meal Cost in Hong Kong. The mean cost of a meal for two in a mid-range restaurant
in Tokyo is $40 (Numbeo.com website). How do prices for comparable meals in Hong Kong compare? The file HongKongMeals contains the costs for a sample of 42 recent
meals for two in Hong Kong mid-range restaurants. a. With 95% confidence, what is the margin of error?
b. What is the 95% confidence interval estimate of the population mean? c. How do prices for meals for two in mid-range restaurants in Hong Kong compare to
prices for comparable meals in Tokyo
?
Data is Meal Price ($)
22.78
33.89
22.77
18.04
23.29
35.28
42.38
36.88
38.55
41.68
25.73
34.19
31.75
25.24
26.32
19.57
36.57
32.97
36.83
30.17
37.29
25.37
24.71
28.79
32.83
43.00
35.23
34.76
33.06
27.73
31.89
38.47
39.42
40.72
43.92
36.51
45.25
33.51
29.17
30.54
26.74
37.93
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Part (a)
Copy and paste the data into a spreadsheet.
Let's say the data values span from cells A1 to A42.
In cell B1, type the formula =STDEV(A:A)
This computes the sample standard deviation of the list of numbers in column A.
Make sure that the equal sign is typed first so the command is executed. Without the equal sign up front, the spreadsheet will display the characters as text (rather than execute the command).
You could type in =STDEV(A1:A42) to be more narrow in that you just want to focus on cells A1 to A42.
However, the first version is better if- you don't know how many items are in the list, and/or
- you want to insert or remove an item from the list (i.e. you want the list to be dynamically sized)
Be careful to NOT put any other unwanted numbers in column A if you use the first command. This column is for the input data only.
I've seen many students make the mistake of sticking calculations at the bottom of the data list, and this will throw off the calculation we want.
Place the calculations off to the side somewhere.
Once again, we can type either
=STDEV(A:A)
or
=STDEV(A1:A42)
for cell B1. The first version is preferred.
The result for cell B1 should be roughly 6.82709987351341
That number is a bit long and messy.
We'll use the ROUND command to make the number appear nicer.
In cell B2, type =ROUND(B1,4) which will round the number in B1 to 4 decimal places. Change the "4" to something larger if you require more precision.
The result of B2 is 6.8271 when rounding B1 to 4 decimal places.
This is the approximate sample standard deviation.
We know the sample size is n = 42 because your teacher gave that info. But what if this wasn't given and all we had was the list of data?
If you want to find out how many numbers are in a column, you can use the COUNT function.
Type =COUNT(A:A) into cell B3 and the number 42 should show up.
Now we need the critical z value.
In cell C1, type 0.95 which is the given confidence level.
In cell C2, type =(1-C1)/2 and the result should be 0.025
In cell C3, type =NORMINV(C2,0,1) and the result is the messy number -1.95996398454005
In cell C4, type =ROUND(C3,3) to get -1.960 or -1.96; we'll drop the negative sign to get z = 1.960 as the critical z value.
Refer to this page to see another method of determining the critical z value.
https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1201518.html
Here are the inputs we'll need:
s = 6.8271 = sample standard deviation
n = 42 = sample size
z = 1.960 = critical z value based on confidence level
Then,
E = margin of error for the mean
E = z*s/sqrt(n)
E = 1.960*6.8271/sqrt(42)
E = 2.06475102503135
E = 2.065
The spreadsheet can be used as a calculator.
Make sure the calculation starts with an equal sign.
So for instance, you could have =1.960*6.8271/sqrt(42) for cell D1.
Then for cell D2, you could have =ROUND(D1,3) to have 2.065 show up in D2.
Answer: 2.065 (approximate)
Round this however your teacher instructs.
-------------------------
Part (b)
We'll use the same spreadsheet we made for part (a)
In cell B4, type =AVERAGE(A:A) to find the arithmetic mean of the data values in column A.
The result for cell B4 should be 32.66 exactly without any rounding done to it.
xbar = 32.66 = sample mean
E = 2.065 = approximate margin of error (see part (a))
L = lower boundary
L = xbar - E
L = 32.66 - 2.065
L = 30.595
L = 30.60
and
U = upper boundary
U = xbar + E
U = 32.66 + 2.065
U = 34.725
U = 34.73
The 95% confidence interval for mu is roughly 30.60 < mu < 34.73
It is in the format L < mu < U
mu = population mean price
A more compact version is of the format (L, U) so it would be (30.60, 34.73)
This more compact version has the drawback we don't know what parameter we're trying to measure unless other context is mentioned.
Interpretation: We are 95% confident the population mean cost of two meals in Hong Kong (mid-range restaurant) is somewhere between $30.60 and $34.73
-------------------------
Part (c)
We're told the mean cost for Tokyo is $40
This is not in the previously calculated confidence interval 30.60 < mu < 34.73 (for Hong Kong)
The 40 is to the right of the confidence interval 30.60 < mu < 34.73
Therefore, we can conclude with 95% confidence that the Tokyo average price is likely larger than the Hong Kong average price.
In other words, it appears to be cheaper to dine in Hong Kong.
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