A producer of chocolate candy bars has developed an improved recipe for one of their most popular brands. They are conducting taste tests using randomly selected mall shoppers. One hundred shoppers were given the original recipe bar to taste and to rate it on a scale of 1 to 10. Another hundred shoppers were given the new recipe candy bar to taste and to rate it on a scale of 1 to 10. The results are in the table below for statistical analysis of variability between the two samples. All of these analyses are run wtih a 95% level of confidence.RATING, 1 to 10Original RecipeNew RecipeSample Mean7.88.1Sample Standard Deviation1.21.6Use this information to answer the following questions.(a) Find the difference between these sample means.(b) Find the variance of the original recipe sample.(c) Find the variance of the new recipe sample.(d) Which test should be run to determine if there is evidence of a difference in consumer preferences between the two recipes? (i) A pooled variance t-test for the difference between two means. (ii) A t-test for the difference between two means assuming unequal variances. (iii) A paired t-test.

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Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.5: Comparing Sets Of Data

Problem 1GP

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A producer of chocolate candy bars has developed an improved recipe for one of their most popular brands. They are conducting taste tests using randomly selected mall shoppers. One hundred shoppers were given the original recipe bar to taste and to rate it on a scale of 1 to 10. Another hundred shoppers were given the new recipe candy bar to taste and to rate it on a scale of 1 to 10. The results are in the table below for statistical analysis of variability between the two samples. All of these analyses are run wtih a 95% level of confidence.

RATING, 1 to 10	Original Recipe	New Recipe
Sample Mean	7.8	8.1
Sample Standard Deviation	1.2	1.6

Use this information to answer the following questions.

(a) Find the difference between these sample means.

(b) Find the variance of the original recipe sample.

(d) Which test should be run to determine if there is evidence of a difference in consumer preferences between the two recipes?

(i) A pooled variance t-test for the difference between two means.

(ii) A t-test for the difference between two means assuming unequal variances.

(iii) A paired t-test.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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