A grocery store sells four different sizes of a popular brand of corn flakes. For the past few years the proportion of boxes they sell of each size has been quite stable: 10% Small, 15% Medium, 60% Large, and 15% Jumbo. They decide to change the pricing of the four sizes and want to see if this changes the proportion of boxes they sell of each size. To test this, a few weeks after changing the prices they take a simple random sample of 120 transactions involving corn flakes and count how many boxes of each size were sold. Here are the results:

Observed number of boxes sold for each box size
Small Medium Large Jumbo
8 24 61 27

Required:
a. We wish to carry out a test of significance to see if the distribution of sizes of cereal boxes soldhas changed. State the null and alternative hypotheses for this test.
b. Find the expected counts for each size box under the assumption that the null hypothesis is true.

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MrRoyal MrRoyal · Answer 1 · 2022-03-31T15:12:03+02:00

The sale of the four different sizes is an illustration of the Goodness of Fit Tests

The expected counts for each size box are 0.8, 3.6, 36.6 and 4.05 respectively

The null hypothesis is always represented by the equality sign i.e. 0 change or no change.

While the alternate hypothesis is always represented by an inequality.

So, the null and the alternate hypotheses are:

Null hypothesis; H0 : The distribution of sizes of all boxes sold of this brand of cereal did not change when the prices changed.
Alternate hypothesis; Ha: The distribution of sizes of all boxes sold of this brand of cereal changed when the prices changed.

This calculated as:

E(x) = np

So, we have:

Small = 8 * 10% = 0.8

Medium = 24 * 15% = 3.6

Large = 61 * 60% = 36.6

Jumbo = 27 * 15% = 4.05

Hence, the expected counts for each size box are 0.8, 3.6, 36.6 and 4.05 respectively