Question 1087484
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Null Hypothesis Using Symbols


H0: *[Tex \LARGE \mu_1 = \mu_2], which is equivalent to *[Tex \LARGE \mu_1 - \mu_2 = 0]


Alternative Hypothesis Using Symbols


H1: *[Tex \LARGE \mu_1 \neq \mu_2], which is equivalent to *[Tex \LARGE \mu_1 - \mu_2 \neq 0]


The symbol *[Tex \LARGE \mu] 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 *[Tex \Large n_1 = 7], *[Tex \Large \overline{x}_1 = 280], and *[Tex \Large s_1 = 21] (to represent the sample size, sample mean, and sample standard deviation in that order)


For chain B we are told *[Tex \Large n_2 = 8], *[Tex \Large \overline{x}_2 = 315], and *[Tex \Large s_2 = 27] (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)


*[Tex \LARGE SE = \sqrt{\frac{\left(s_1\right)^2}{n_1}+\frac{\left(s_2\right)^2}{n_2}}]


*[Tex \LARGE SE = \sqrt{\frac{\left(21\right)^2}{7}+\frac{\left(27\right)^2}{8}}]


*[Tex \LARGE SE = \sqrt{\frac{441}{7}+\frac{729}{8}}]


*[Tex \LARGE SE = \sqrt{63+91.125]


*[Tex \LARGE SE = \sqrt{154.125]


*[Tex \LARGE SE \approx 12.4147090179351]


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With the SE, we can find the T test statistic


I'll call this variable *[Tex \LARGE T_s] (T sub s) to differentiate it from the T critical values later on.


*[Tex \LARGE T_s = \frac{\overline{x}_1-\overline{x}_2}{SE}]


*[Tex \LARGE T_s \approx \frac{280-315}{12.4147090179351}]


*[Tex \LARGE T_s \approx \frac{-35}{12.4147090179351}]


*[Tex \LARGE T_s \approx -2.81923643554083]


*[Tex \LARGE T_s \approx -2.82]


The T test statistic is approximately -2.82


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The T critical values, which I'll call *[Tex \LARGE T_L] and *[Tex \LARGE T_R] (for the left and right critical values respectively) are approximately


*[Tex \LARGE T_L \approx -2.160]


*[Tex \LARGE T_R \approx 2.160]


Use a table or calculator to find these critical values. In my case, I used <a href="http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf">this table</a> 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 <a href="http://www.statsdirect.com/help/parametric_methods/utt.htm">this page</a> for further reading. 


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To recap so far, we found the following


*[Tex \LARGE T_s \approx -2.82] which is the T statistic


*[Tex \LARGE T_L \approx -2.160] which is the left critical T value


*[Tex \LARGE T_R \approx 2.160] which is the right critical T value


It's clear that *[Tex \LARGE T_s] is NOT between *[Tex \LARGE T_L] and *[Tex \LARGE T_R]. So we are not in the noncritical region. The test statistic is in the critical region. 


This visual shows *[Tex \LARGE T_s] insinde the red rejection or critical region and outside of the blue noncritical region. The critical region is formed by the fenceposts *[Tex \LARGE T_L] and *[Tex \LARGE T_R]
<img src = "https://i.imgur.com/44udOk0.png">
(Image generated by <a href = "https://www.geogebra.org/home?ggbLang=en">GeoGebra</a> 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.</font>