Question 1087484: A reporter bought hamburgers at randomly selected stores of two different restaurant chains, and had the number of Calories in each hamburger measured. Can the reporter conclude, at alpha = 0.05, that the hamburgers from the two chains have a different number of Calories??
Chain A sample size 7, sample mean 280 cal, sample standard deviation 21 cal
Chain B sample size 8, sample mean 315 cal, sample standard deviation 27 cal
a. No, because the test value –0.23 is inside the noncritical region
b. Yes, because the test value –0.23 is inside the noncritical region
c. Yes, because the test value –2.82 is outside the noncritical region
d. No, because the test value –1.29 is inside the noncritical region
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Null Hypothesis Using Symbols
H0:  , which is equivalent to 
Alternative Hypothesis Using Symbols
H1:  , which is equivalent to 
The symbol  is the greek letter mu.
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Null Hypothesis in English: The two hamburger chains have burgers with the same number of Calories
Alternative Hypothesis in English: The two hamburger chains have burgers with a different number of Calories
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This is a two tailed test. We're going to use a two sample unpaired T test to test the hypothesis.
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Given Information:
For chain A we have  ,  , and  (to represent the sample size, sample mean, and sample standard deviation in that order)
For chain B we are told  ,  , and  (again representing the sample size, sample mean, and sample standard deviation in that order but this time we have '2's attached to the variable's instead of '1's to indicate a different label)
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Using the info about the sample sizes and sample standard deviations, let's compute the Standard Error (SE)
^2}{n_1}+\frac{\left(s_2\right)^2}{n_2}})
^2}{7}+\frac{\left(27\right)^2}{8}})
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With the SE, we can find the T test statistic
I'll call this variable  (T sub s) to differentiate it from the T critical values later on.
The T test statistic is approximately -2.82
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The T critical values, which I'll call  and  (for the left and right critical values respectively) are approximately
Use a table or calculator to find these critical values. In my case, I used this table to find the critical values. How? By looking in the df = 13 row (df = n1+n2-2 = 7+8-2 = 15-2 = 13; see note below). Then look in the column that has "two tails = 0.05" to find 2.160 in the table. This means that P(-2.160 < T < 2.160) is roughly equal to 0.95 and the area of 0.05 is in the tails (0.025 in each individual tail)
Recall that alpha = 0.05 is the significance level in this case.
Note: I'm assuming the population variances are equal. This makes the df value much easier to compute. If they were assumed to be unequal, then we'd have to use a nasty formula to compute the df. I checked both versions of the df and got roughly 12.855898757298 when variances were assumed to be unequal, which is close enough to 13 in my opinion. See this page for further reading.
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To recap so far, we found the following
which is the T statistic
which is the left critical T value
which is the right critical T value
It's clear that is NOT between and . So we are not in the noncritical region. The test statistic is in the critical region.
This visual shows insinde the red rejection or critical region and outside of the blue noncritical region. The critical region is formed by the fenceposts and
(Image generated by GeoGebra which is free graphing software)
The decision is therefore to reject the null hypothesis (H0). So we accept the alternate hypothesis (H1).
The conclusion, translated to common english, is that the burgers do have different calorie counts between the two burger chains.
So the short answer is "yes, the reporter can conclude that the two chains have different number of calories (at alpha = 0.05 significance level)"
The final answer is choice C.
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