Answer to Question #174140 in Statistics and Probability for Denisse Bisuña

Question

find the point estimate of the population parameter μ, and the standard deviation for each of the following sets of data. show your computations.

weights (in grams) of packed ground coffee:

350 346 350 346 350 348 351 351

340 347 344 340 340 340 345 347

355 348 351 355 347 352 356 352

347 348 347 347 347 347 346 347

348 351 347 348 348 348 348 349

348 349 348 348 349 348 347 349

349 349 349 349 349 348 346 349

time (in seconds) it took Lydia to finish a 100m practice race:

15 12 16 12 15 15 15 16

14 13 14 14 16 14 14 16

12 12 12 13 12 15 12 13

12 15 15 13 12 12 12 12

15 11 15 15 15 15 15 15

18 16 17 16 15 16 16 18

18 17 18 16 15 14 18 16

Percentage of children who watch tv before bedtime:

70 67 58 60 69 69 70 62

69 59 77 59 52 79 59 59

80 42 60 59 68 40 68 68

56 66 60 40 57 57 70 71

72 54 52 67 62 59 71 72

81 49 45 78 78 69 68 69

Percentage of parents in favor of including cultural values in the mathematics curriculum:

90 70 80 76 81 82 76 84

89 59 76 78 75 89 79 89

92 42 58 84 75 90 80 78

82 68 82 82 68 78 79 80

72 54 83 80 78 79 80 84

81 69 78 78 80 82 81 90

Answer 1

The sample mean $(\bar{x})$ is a point estimate of the population mean, $\mu.$

The sample variance $(s^2)$ is a point estimate of the population variance $(\sigma^2).$

The sample standard deviation $(s)$ is a point estimate of the population standard deviation $(\sigma).$

1.

$$\bar{x}=\dfrac{1}{n}\sum_ix_i=\dfrac{19488}{56}=348(g)$$

$$s^2=\dfrac{\sum_i(x_i-\bar{x})^2}{n-1}$$

$$=\dfrac{(350-348)^2+(346-348)^2+...+(349-348)^2}{56-1}$$

$$=\dfrac{560}{55}=\dfrac{112}{11}\approx10.181818$$

$$s=\sqrt{s^2}=\sqrt{\dfrac{112}{11}}\approx3.191(g)$$

2.

$$\bar{x}=\dfrac{1}{n}\sum_ix_i=\dfrac{815}{56}\approx14.554(s)$$

$$s^2=\dfrac{\sum_i(x_i-\bar{x})^2}{n-1}$$

$$=\dfrac{(15-\dfrac{815}{56})^2+...+(12-\dfrac{815}{56})^2}{56-1}$$

$$\approx3.597078$$

$$s=\sqrt{s^2}\approx\sqrt{3.597078}\approx1.897(s)$$

3.

$$\bar{x}=\dfrac{1}{n}\sum_ix_i=\dfrac{3046}{48}=\dfrac{1523}{24}\approx63.458(\%)$$

$$s^2=\dfrac{\sum_i(x_i-\bar{x})^2}{n-1}$$

$$=\dfrac{(70-\dfrac{1523}{24})^2+...+(69-\dfrac{1523}{24})^2}{48-1}$$

$$\approx103.742908$$

$$s=\sqrt{s^2}\approx10.185(\%)$$

4.

$$\bar{x}=\dfrac{1}{n}\sum_ix_i=\dfrac{3740}{48}=\dfrac{935}{12}\approx77.917(\%)$$

$$s^2=\dfrac{\sum_i(x_i-\bar{x})^2}{n-1}$$

$$=\dfrac{(90-\dfrac{935}{12})^2+...+(90-\dfrac{935}{12})^2}{48-1}$$

$$\approx91.524823$$

$$s=\sqrt{s^2}\approx9.567(\%)$$