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joja [24]
2 years ago
8

2. In your own words, explain HOW to find the standard deviation of this data set. (20 points) 3. What does the standard deviati

on tell you about this data? (20 points) I NEEDDDDD THISSSSSS PLSSSSSSSSSSSSSSSS
Mathematics
1 answer:
Ket [755]2 years ago
8 0

Answer:

\sigma = 11.28 --- Standard deviation

Step-by-step explanation:

Given

See attachment for graph

Solving (a): Explain how the standard deviation is calculated.

<u>Start by calculating the mean</u>

To do this, we divide the sum of the products of grade and number of students by the total number of students;

i.e.

\bar x = \frac{\sum fx}{\sum f}

So, we have:

\bar x = \frac{50 * 1 + 57 * 2 + 60 * 4 + 65 * 3 +72 *3 + 75 * 12 + 77 * 10 + 81 * 6 + 83 * 6 + 88 * 9 + 90 * 12 + 92 * 12 +95 * 2 + 99 * 4 + 100 * 5}{1 + 2 + 4 + 3 +3 + 12 + 10 + 6 + 6 + 9 + 12 + 12 +2 + 4 + 5}

\bar x = \frac{7531}{91}

\bar x = 82.76

Next, calculate the variance using the following formula:

\sigma^2 = \frac{\sum f(x - \bar x)^2}{\sum f}

i.e subtract the mean from each dataset; take the squares; add up the squares; then divide the sum by the number of dataset

So, we have:

\sigma^2 = \frac{1 * (50 - 82.76)^2 +...................+ 5 * (100 - 82.76)^2}{91}

\sigma^2 = \frac{11580.6816}{91}

\sigma^2 = 127.26

Lastly, take the square root of the variance to get the standard deviation

\sigma = \sqrt{\sigma^2}

\sigma = \sqrt{127.26}

\sigma = 11.28 --- approximated

<em>Hence, the standard deviation is approximately 11.28</em>

Considering the calculated mean (i.e. 82.76), the standard deviation (i.e. 11.28) is small and this means that the grade of the students are close to the average grade.

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Suppose the standard deviation of a normal population is known to be 3, and H0 asserts that the mean is equal to 12. A random sa
blagie [28]

Answer:

z=\frac{12.95-12}{\frac{3}{\sqrt{36}}}=1.9  

p_v =P(z>1.9)=0.0287  

If we compare the p value and the significance level given \alpha=0.05 we see that p_v so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the true mean is higher than 12 at 5% of signficance.  

Step-by-step explanation:

Data given and notation  

\bar X=12.95 represent the sample mean

\sigma=3 represent the population standard deviation for the sample  

n=36 sample size  

\mu_o =12 represent the value that we want to test  

\alpha=0.05 represent the significance level for the hypothesis test.  

z would represent the statistic (variable of interest)  

p_v represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the mean is equal to 12, the system of hypothesis would be:  

Null hypothesis:\mu \leq 12  

Alternative hypothesis:\mu > 12  

Since we know the population deviation, is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:  

z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}} (1)  

z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic  

We can replace in formula (1) the info given like this:  

z=\frac{12.95-12}{\frac{3}{\sqrt{36}}}=1.9  

P-value  

Since is a right tailed test the p value would be:  

p_v =P(z>1.9)=0.0287  

Conclusion  

If we compare the p value and the significance level given \alpha=0.05 we see that p_v so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the true mean is higher than 12 at 5% of signficance.  

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