Math  /  Data & Statistics

QuestionYou are conducting a multinomial hypothesis test ( α=0.05\alpha=0.05 ) for the claim that all 5 categories are equally likely to be selected. Complete the table. \begin{tabular}{|c|c|c|c|} \hline Category & Observed Frequency & Expected Frequency & (OE)2E\frac{(O-E)^{2}}{E} \\ \hline A & 20 & & \\ \hline B & 13 & & \\ \hline C & 12 & & \\ \hline D & 8 & & \\ \hline E & 24 & & \\ \hline \end{tabular}
Report all answers accurate to three decimal places. But retain unrounded numbers for future

Studdy Solution

STEP 1

What is this asking? We need to fill in the missing values in the table to perform a multinomial hypothesis test, checking if each of the five categories is equally likely to be chosen. Watch out! Don't forget that rounding too early can throw off your final result!
Keep those unrounded numbers handy until the very end.

STEP 2

1. Calculate the total observed frequency.
2. Calculate the expected frequency for each category.
3. Calculate the chi-squared components for each category.

STEP 3

Let's **add up** all the observed frequencies!
This gives us the **total number** of observations across all categories.
We've got 2020 for A, 1313 for B, 1212 for C, 88 for D, and 2424 for E.

STEP 4

So, the **total observed frequency** is 20+13+12+8+24=7720 + 13 + 12 + 8 + 24 = 77.

STEP 5

Since we're testing if all categories are *equally likely*, the **expected frequency** for each category is simply the **total observed frequency** divided by the **number of categories**.

STEP 6

We have 55 categories and a **total observed frequency** of 7777, so the **expected frequency** for *each* category is 775=15.4\frac{77}{5} = 15.4.

STEP 7

Now for the fun part!
For each category, we'll calculate (OE)2E\frac{(O-E)^{2}}{E}, where OO is the **observed frequency** and EE is the **expected frequency**.
This tells us how much each category contributes to the overall chi-squared statistic.

STEP 8

(2015.4)215.4=(4.6)215.4=21.1615.41.374\frac{(20 - 15.4)^{2}}{15.4} = \frac{(4.6)^{2}}{15.4} = \frac{21.16}{15.4} \approx 1.374

STEP 9

(1315.4)215.4=(2.4)215.4=5.7615.40.374\frac{(13 - 15.4)^{2}}{15.4} = \frac{(-2.4)^{2}}{15.4} = \frac{5.76}{15.4} \approx 0.374

STEP 10

(1215.4)215.4=(3.4)215.4=11.5615.40.751\frac{(12 - 15.4)^{2}}{15.4} = \frac{(-3.4)^{2}}{15.4} = \frac{11.56}{15.4} \approx 0.751

STEP 11

(815.4)215.4=(7.4)215.4=54.7615.43.556\frac{(8 - 15.4)^{2}}{15.4} = \frac{(-7.4)^{2}}{15.4} = \frac{54.76}{15.4} \approx 3.556

STEP 12

(2415.4)215.4=(8.6)215.4=73.9615.44.803\frac{(24 - 15.4)^{2}}{15.4} = \frac{(8.6)^{2}}{15.4} = \frac{73.96}{15.4} \approx 4.803

STEP 13

Here's the completed table with all the values we calculated, rounded to three decimal places:
| Category | Observed Frequency | Expected Frequency | (OE)2E\frac{(O-E)^{2}}{E} | |---|---|---|---| | A | 2020 | 15.415.4 | 1.3741.374 | | B | 1313 | 15.415.4 | 0.3740.374 | | C | 1212 | 15.415.4 | 0.7510.751 | | D | 88 | 15.415.4 | 3.5563.556 | | E | 2424 | 15.415.4 | 4.8034.803 |

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