Measurements of the sodium content in samples of two brands of chocolate bar yield the following results (in grams) : Brand A: 34.36 31.26 37.36 28.52 33.14 32.74 34.34 34.33 34.95Brand B: 41.08 38.22 39.59 38.82 36.24 37.73 35.03 39.22 34.13 34.33 34.98 29.64 40.60 Let formula415. mml represent the population mean for Brand B and let formula417. mml represent the population mean for Brand A. Find a 98% confidence interval for the difference formula419. mml. Round down the degrees of freedom to the nearest integer and round the answers to three decimal places.

Question

Jenkins

Mathematics

12 December, 16:19

Measurements of the sodium content in samples of two brands of chocolate bar yield the following results (in grams) : Brand A: 34.36 31.26 37.36 28.52 33.14 32.74 34.34 34.33 34.95Brand B: 41.08 38.22 39.59 38.82 36.24 37.73 35.03 39.22 34.13 34.33 34.98 29.64 40.60 Let formula415. mml represent the population mean for Brand B and let formula417. mml represent the population mean for Brand A. Find a 98% confidence interval for the difference formula419. mml. Round down the degrees of freedom to the nearest integer and round the answers to three decimal places.

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Answers (1)

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Case Todd · Answer 1

Step-by-step explanation:

For brand A,

Mean, x1 = (34.36 + 31.26 + 37.36 + 28.52 + 33.14 + 32.74 + 34.34 + 34.33 + 34.95) / 9 = 33.44

standard deviation, s1 = √ (summation (x - mean) ²/n

Summation (x - mean) ² = (34.36 - 33.44) ^2 + (31.26 - 33.44) ^2 + (37.36 - 33.44) ^2 + (28.52 - 33.44) ^2 + (33.14 - 33.44) ^2 + (32.74 - 33.44) ^2 + (34.34 - 33.44) ^2 + (34.33 - 33.44) ^2 + (34.95 - 33.44) ^2 = 49.6338

Standard deviation = √ (49.6338/9

s1 = 2.35

For brand B,

Mean, x2 = (41.08 + 38.22 + 39.59 + 38.82 + 36.24 + 37.73 + 35.03 + 39.22 + 34.13 + 34.33 + 34.98 + 29.64 + 40.60) / 13 = 36.89

Summation (x - mean) ² = (41.08 - 36.89) ^2 + (38.22 - 36.89) ^2 + (39.59 - 36.89) ^2 + (38.82 - 36.89) ^2 + (36.24 - 36.89) ^2 + (37.73 - 36.89) ^2 + (35.03 - 36.89) ^2 + (39.22 - 36.89) ^2 + (34.13 - 36.89) ^2 + (34.33 - 36.89) ^2 + (34.98 - 36.89) ^2 + (29.64 - 36.89) ^2 + (40.60 - 36.89) ^2 = 124.5024

Standard deviation = √ (124.5024/13

s2 = 3.09

The formula for determining the confidence interval for the difference of two population means is expressed as

Confidence interval = (x1 - x2) ± z√ (s²/n1 + s2²/n2)

For a 98% confidence interval, we would determine the z score from the t distribution table because the number of samples are small.

Degree of freedom =

(n1 - 1) + (n2 - 1) = (9 - 1) + (13 - 1) = 20

z = 2.528

x1 - x2 = 33.44 - 36.89 = - 3.45

Margin of error = z√ (s1²/n1 + s2²/n2) = 2.528√ (2.35²/9 + 3.09²/13) = 2.935

The 98% confidence interval is

- 3.45 ± 2.935