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DATA
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Question
The mean cost of a meal for two in a mid-range restaurant in Tokyo is $40 (Numbeo website, December 14, 2014). How do prices for comparable meals in Hong Kong compare? The DATAfile Hong Kong Meals 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 (to 2 decimals)?
b. What is the 95% confidence interval estimate of the population mean (to 2 decimals)?
c. How do prices for meals for two in mid-range restaurants in Hong Kong compare to prices for comparable meals in Tokyo restaurants?
Answer:
a
The margin of error is [tex]E = 2.13 [/tex]
b
The 95% confidence interval is [tex] 30.53 < \mu < 34.79 [/tex]
c
The cost of mean in Hong Kong is lower
Step-by-step explanation:
From the question we are told that
The mean cost of a meal for two in a mid-range restaurant in Tokyo is [tex]\mu_1 = \$ 40[/tex]
Generally the mean for a meal in Hong Kong is mathematically represented as
[tex]\= x = \frac{\sum x_i }{n}[/tex]
=> [tex]\= x = \frac{ 22.78+ 33.89 +\cdots + 37.93 }{42}[/tex]
=> [tex]\= x =32.66[/tex]
Generally the standard deviation is mathematically represented as
[tex]\sigma = \sqrt{\frac{ \sum (x_i - \= x )^2}{n-1} }[/tex]
=> [tex]\sigma = \sqrt{\frac{ (22.78 - 32.66 )^2+ (33.89 - 32.66 )^2+\cdots + (37.93 - 32.66 )^2}{42-1} }[/tex]
=> [tex]\sigma = 6.83[/tex]
Generally the degree of freedom is mathematically represented as
[tex]df = n- 1[/tex]
=> [tex]df = 42 - 1[/tex]
=> [tex]df = 41[/tex]
From the question we are told the confidence level is 95% , hence the level of significance is
[tex]\alpha = (100 - 95 ) \%[/tex]
=> [tex]\alpha = 0.05[/tex]
Generally from the t distribution table the critical value of [tex]\frac{\alpha }{2}[/tex] at a degree of freedom of [tex]df = 41[/tex] is
[tex]t_{\frac{\alpha }{2} , 41} = 2.020 [/tex]
Generally the margin of error is mathematically represented as
[tex]E = t_{\frac{\alpha }{2} ,41 } * \frac{\sigma }{\sqrt{n} }[/tex]
=> [tex]E = 2.0203 * \frac{6.83 }{\sqrt{42} }[/tex]
=> [tex]E = 2.13 [/tex]
Generally 95% confidence interval is mathematically represented as
[tex]\= x -E < \mu < \=x +E[/tex]
=> [tex] 32.66 - 2.13 < \mu < 32.66 + 2.13 [/tex]
=> [tex] 30.53 < \mu < 34.79 [/tex]
Generally comparing the mean cost of a meal for two in a mid-range restaurant in Tokyo to the 95% confidence interval estimate of the population mean cost of mean in Hong Kong we see that the cost of meals in Hong Kong is lower.
