Cereals Supermarkets often place similar types of cerealon the same supermarket shelf. We have data on theshelf as well as the sugar, sodium, and calorie content of77 cereals. Does sugar content vary by shelf? At the topof the next column is a boxplot and an ANOVA table forthe 77 cereals.Sum ofSquaresMeanSquare F-Ratio P-ValueShelf 2 248.4079 124.204 7.3345 0.0012Error 74 1253.1246 16.934Total 76 1501.5325Means and Std DeviationsLevel n Mean StdDev1 20 4.80000 4.572232 21 9.61905 4.128883 36 6.52778 3.83582a) What are the null and alternative hypotheses?b) What does the ANOVA table say about the nullhypothesis? (Be sure to report this in terms of Sugarsand Shelves.)c) Can we conclude that cereals on shelf 2 have a highermean sugar content than cereals on shelf 3? Can weconclude that cereals on shelf 2 have a higher meansugar content than cereals on shelf 1? What can weconclude?d) To check for significant differences between the shelfmeans, we can use a Bonferroni test, whose results areshown below. For each pair of shelves, the differenceis shown along with its standard error and significancelevel. What does it say about the questions in part c?Dependent Variable: SUGARS(I)SHELF(J)SHELFMeanDifference(I-J)Std.Error P-Value95%ConfidenceIntervalBonferroni LowerBoundUpperBound1 2 -4.819(*) 1.2857 0.001 -7.969 -1.6703 -1.728 1.1476 0.409 -4.539 1.0842 1 4.819(*) 1.2857 0.001 1.670 7.9693 3.091(*) 1.1299 0.023 0.323 5.8593 1 1.728 1.1476 0.409 -1.084 4.5392 -3.091(*) 1.1299 0.023 -5.859 -0.323
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Cereals Supermarkets often place similar types of cereal
on the same supermarket shelf. We have data on the
shelf as well as the sugar, sodium, and calorie content of
77 cereals. Does sugar content vary by shelf? At the top
of the next column is a boxplot and an ANOVA table for
the 77 cereals.Sum of
Squares
Mean
Square F-Ratio P-Value
Shelf 2 248.4079 124.204 7.3345 0.0012
Error 74 1253.1246 16.934
Total 76 1501.5325
Means and Std Deviations
Level n Mean StdDev
1 20 4.80000 4.57223
2 21 9.61905 4.12888
3 36 6.52778 3.83582
a) What are the null and alternative hypotheses?
b) What does the ANOVA table say about the null
hypothesis? (Be sure to report this in terms of Sugars
and Shelves.)
c) Can we conclude that cereals on shelf 2 have a higher
mean sugar content than cereals on shelf 3? Can we
conclude that cereals on shelf 2 have a higher mean
sugar content than cereals on shelf 1? What can we
conclude?
d) To check for significant differences between the shelf
means, we can use a Bonferroni test, whose results are
shown below. For each pair of shelves, the difference
is shown along with its standard error and significance
level. What does it say about the questions in part c?
Dependent Variable: SUGARS
(I)
SHELF
(J)
SHELF
Mean
Difference
(I-J)
Std.
Error P-Value
95%
Confidence
Interval
Bonferroni Lower
Bound
Upper
Bound
1 2 -4.819(*) 1.2857 0.001 -7.969 -1.670
3 -1.728 1.1476 0.409 -4.539 1.084
2 1 4.819(*) 1.2857 0.001 1.670 7.969
3 3.091(*) 1.1299 0.023 0.323 5.859
3 1 1.728 1.1476 0.409 -1.084 4.539
2 -3.091(*) 1.1299 0.023 -5.859 -0.323
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