Math 103 Quiz 4 Chapter 8 2021c ✓ Solved

```html

Math 103 quiz 4 chapter 8 2021c Upload file(s) as "Last

Math 103 quiz 4 chapter 8 2021c. Open book & notes, can use calculating device, formula sheets + extra paper. Use formula sheets including Normal distribution (z) tables. If completing problems manually, show work for credit.

1. Use this data set { 65, 38, 53, 62, 44, 87, 27, 78, 21, 18, 60, 78, 2, 44 } and find the:

a) median

b) mode ( Round answers to a few decimals (1 or 2) Show work if NOT using your calculator)

c) mean

d) sample standard deviation

Bonus 1 (+0.1%) use the above data & find the Range... (+0.1%) & Midrange.

2. a) Use the data below to make a scatter plot. Hint: count gridlines (blocks) by 1→ x 5 2 1.5 5..4 y .5 1.3 2.7. What kind of linear correlation seems to exist:

b) positive, negative, or none?

3. A fish species has normally distributed weights with standard deviation 2.9 and mean = 22.5 lbs. Show these 3 parts for each:

a) Fill in the appropriate formula & show the calculated z.

b) Next, round it up (or down) to the closest z-value (2 decimals) listed in the N-distribution tables, and write that value.

c) Find the # using the table (show all 4 digits, % or decimal).

i) What is the percentage (%) of data less than 22?

ii) what is the % of data greater than 26?

iii) what is the % of data between 21 and 28.5?

Paper For Above Instructions

The quiz involves several statistical concepts, starting with basic data analysis tasks such as finding the median, mode, mean, and sample standard deviation of a given dataset. This dataset includes the following values:

  • 65, 38, 53, 62, 44, 87, 27, 78, 21, 18, 60, 78, 2, 44.

To calculate the median, we first need to order the data from least to greatest:

  • 2, 18, 21, 27, 38, 44, 44, 53, 60, 62, 65, 78, 78, 87.

The median is the middle value of the sorted dataset. With 14 values, the median will be the average of the 7th and 8th values:

(44 + 53) / 2 = 48.5

Therefore, the median of the dataset is 48.5.

The mode is the number that appears most frequently in the dataset. From our ordered list, we can see that 44 and 78 each appear twice, making them both modes. Thus, the modes are 44 and 78.

Next, we calculate the mean:

Mean = (Sum of all values) / (Number of values) = (65 + 38 + 53 + 62 + 44 + 87 + 27 + 78 + 21 + 18 + 60 + 78 + 2 + 44) / 14 = 50.0714 (rounded to two decimal places, the mean is 50.07).

Now, let's calculate the sample standard deviation. The formula for the sample standard deviation (s) is given by:

s = sqrt[(Σ(x - x̄)²) / (n - 1)]

Where:

  • Σ is the summation notation.
  • x is a value in the dataset.
  • x̄ is the mean of the dataset.
  • n is the number of values in the dataset.

Calculating each deviation from the mean, squaring those values, and summing gives us:

Σ(x - x̄)² = (65 - 50.07)² + (38 - 50.07)² + ... + (44 - 50.07)².

After going through this process, we find that σ² ≈ 757.59, therefore:

s = sqrt(757.59 / (14 - 1)) ≈ sqrt(58.09) ≈ 7.62.

The sample standard deviation is approximately 7.62.

Next, we calculate the bonus values: the range is the difference between the maximum and minimum values:

Range = Max - Min = 87 - 2 = 85.

The midrange is calculated as:

Midrange = (Max + Min) / 2 = (87 + 2) / 2 = 44.5.

Moving on to question two's scatter plot, we have the following values:

  • Data points for x: 5, 2, 1.5, 5, 4
  • Data points for y: 0.5, 1.3, 2.7

When plotting these points, we find a visual inspection indicates a positive correlation, as y generally increases with x.

For question three regarding the fish species weights, we have a mean weight of 22.5 lbs with a standard deviation of 2.9 lbs. To find the z-scores for various weights, we use the formula:

z = (x - μ) / σ

For data less than 22 lbs:

z = (22 - 22.5) / 2.9 ≈ -0.17.

Referring to the z-table, the corresponding area for z = -0.17 is approximately 0.4325, meaning about 43.25% of data is less than 22 lbs.

For weights greater than 26 lbs:

z = (26 - 22.5) / 2.9 ≈ 1.21.

From the z-table, the area for z = 1.21 is approximately 0.8869, therefore:

100% - 88.69% = 11.31% of fish are heavier than 26 lbs.

Lastly, for the percentage of data between 21 and 28.5 lbs:

Calculating both z-scores:

z₁ (21 lbs) = (21 - 22.5) / 2.9 ≈ -0.52.

z₂ (28.5 lbs) = (28.5 - 22.5) / 2.9 ≈ 2.07.

Using the z-table:

Area for z = -0.52 is approximately 0.3015 and for z = 2.07 is approximately 0.9808.

The percentage of data between these values is:

0.9808 - 0.3015 = 0.6793 or about 67.93%.

References

  • Wackerly, D., Mendenhall, W., & Scheaffer, L. D. (2008). Mathematical Statistics with Applications. Brooks/Cole.
  • Moore, D. S. (2013). Introductory Statistics. W. H. Freeman.
  • Field, A. P. (2013). Discovering Statistics Using IBM SPSS Statistics. SAGE Publications.
  • Bluman, A. G. (2017). Elementary Statistics: A Step by Step Approach. McGraw-Hill Education.
  • Ghasemi, A., & Zahediasl, S. (2012). Normality Tests for Statistical Analysis: A Guide for Non-statisticians. International Journal of Endocrinology and Metabolism, 10(2), 486-489.
  • Gravetter, F. J., & Wallnau, L. B. (2014). Statistics for The Behavioral Sciences. Cengage Learning.
  • Glass, G. V., & Hopkins, K. D. (1996). Statistical Methods in Education and Psychology. Allyn and Bacon.
  • Trochim, W. M. K. (2006). The Research Methods Knowledge Base. Atomic Dog Publishing.
  • Hogg, R. V., & Tanis, E. A. (2015). Probability and Statistical Inference. Pearson.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.

```

Related Assignments