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Katen [24]
2 years ago
9

Solid fats are more likely to raise blood cholesterol levels than liquid fats. Suppose a nutritionist analyzed the percentage of

saturated fat for a sample of 6 brands of stick margarine (solid fat) and for a sample of 6 brands of liquid margarine and obtained the following results: Exam Image Exam Image We want to determine if there a significant difference in the average amount of saturated fat in solid and liquid fats. What is the test statistic
Mathematics
1 answer:
Andrews [41]2 years ago
3 0

Answer:

t = 31.29

Step-by-step explanation:

Given

\begin{array}{ccccccc}{Stick} & {25.8} & {26.9} & {26.2} & {25.3} & {26.7}& {26.1} \ \\ {Liquid} & {16.9} & {17.4} & {16.8} & {16.2} & {17.3}& {16.8} \ \end{array}

Required

Determine the test statistic

Let the dataset of stick be A and Liquid be B.

We start by calculating the mean of each dataset;

\bar x =\frac{\sum x}{n}

n, in both datasets in 6

For A

\bar x_A =\frac{25.8+26.9+26.2+25.3+26.7+26.1}{6}

\bar x_A =\frac{157}{6}

\bar x_A =26.17

For B

\bar x_B =\frac{16.9+17.4+16.8+16.2+17.3+16.8}{6}

\bar x_B =\frac{101.4}{6}

\bar x_B =16.9

Next, calculate the sample standard deviation

This is calculated using:

s = \sqrt{\frac{\sum(x - \bar x)^2}{n-1}}

For A

s_A = \sqrt{\frac{\sum(x - \bar x_A)^2}{n-1}}

s_A = \sqrt{\frac{(25.8-26.17)^2+(26.9-26.17)^2+(26.2-26.17)^2+(25.3-26.17)^2+(26.7-26.17)^2+(26.1-26.17)^2}{6-1}}

s_A = \sqrt{\frac{1.7134}{5}}

s_A = \sqrt{0.34268}

s_A = 0.5854  

For B

s_B = \sqrt{\frac{\sum(x - \bar x_B)^2}{n-1}}

s_B = \sqrt{\frac{(16.9 - 16.9)^2+(17.4- 16.9)^2+(16.8- 16.9)^2+(16.2- 16.9)^2+(17.3- 16.9)^2+(16.8- 16.9)^2}{6-1}}

s_B = \sqrt{\frac{0.92}{5}}

s_B = \sqrt{0.184}

s_B = 0.4290

Calculate the pooled variance

S_p^2 = \frac{(n_A - 1)*s_A^2 + (n_B - 1)*s_B^2}{(n_A+n_B-2)}

S_p^2 = \frac{(6 - 1)*0.5854^2 + (6 - 1)*0.4290^2}{(6+6-2)}

S_p^2 = \frac{2.6336708}{10}

S_p^2 = 0.2634

Lastly, calculate the test statistic using:

t = \frac{(\bar x_A - \bar x_B) - (\mu_A - \mu_B)}{\sqrt{S_p^2/n_A +S_p^2/n_B}}

We set

\mu_A = \mu_B

So, we have:

t = \frac{(\bar x_A - \bar x_B) - (\mu_A - \mu_A)}{\sqrt{S_p^2/n_A +S_p^2/n_B}}

t = \frac{(\bar x_A - \bar x_B) }{\sqrt{S_p^2/n_A +S_p^2/n_B}}

The equation becomes

t = \frac{(26.17 - 16.9) }{\sqrt{0.2634/6 +0.2634/6}}

t = \frac{9.27}{\sqrt{0.0878}}

t = \frac{9.27}{0.2963}

t = 31.29

<em>The test statistic is 31.29</em>

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