Question

a group of researchers is studying the effect of different types of light on plant growth. they categorize 70 plants into two categories based on their growth (healthy or unhealthy) and expose them to three different types of light (natural sunlight, led light, fluorescent light). after 4 weeks, they collect the following data:

light type	healthy	unhealthy	total
led light	12	8	20
fluorescent light	8	12	20
total	40	30	70

researchers aim to determine whether the type of light has a significant effect on plant growth. calculate the expected counts and the chi-squared test statistic by calculating the contribution of each cell.

light type	healthy	unhealthy	total
led light	c	d	20
fluorescent light	e	f	20
total	40	30	70

enter the expected counted in the space below.

a: (round your answer to 2 decimal places)
b: (round your answer to 2 decimal places)
c: (round your answer to 2 decimal places)
d: (round your answer to 2 decimal places)
e: (round your answer to 2 decimal places)

Explanation:

🆕 New Concept Discovered: Chi-Square Test of Independence
Finding expected counts in contingency tables

Step 1: Formula for Expected Counts

To find the expected count for each cell in a contingency table under the assumption of independence, we use the formula:

$$ \text{Expected Count} = \frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}} $$

The grand total is \(70\).

Step 2: Calculate Expected Counts for Natural Sunlight (Row 1)

For A (Healthy, Natural Sunlight):

• Row Total = \(30\)
• Column Total = \(40\)

$$ A = \frac{30 \times 40}{70} = \frac{1200}{70} \approx 17.14 $$

For B (Unhealthy, Natural Sunlight):

• Row Total = \(30\)
• Column Total = \(30\)

$$ B = \frac{30 \times 30}{70} = \frac{900}{70} \approx 12.86 $$

Step 3: Calculate Expected Counts for LED Light (Row 2)

For C (Healthy, LED Light):

• Row Total = \(20\)
• Column Total = \(40\)

$$ C = \frac{20 \times 40}{70} = \frac{800}{70} \approx 11.43 $$

For D (Unhealthy, LED Light):

• Row Total = \(20\)
• Column Total = \(30\)

$$ D = \frac{20 \times 30}{70} = \frac{600}{70} \approx 8.57 $$

Step 4: Calculate Expected Count for E (Healthy, Fluorescent Light)

For E (Healthy, Fluorescent Light):

• Row Total = \(20\)
• Column Total = \(40\)

$$ E = \frac{20 \times 40}{70} = \frac{800}{70} \approx 11.43 $$

Answer:

• A: 17.14
• B: 12.86
• C: 11.43
• D: 8.57
• E: 11.43