17.9 - The Chi-squared (x2) Test

Question 1

What is the purpose of the chi-squared (χ²) test in scientific investigations?

Answer 1

• The chi-squared test is used to test the null hypothesis.
• It examines whether the difference between observed and expected results in an investigation is due to chance or if it is statistically significant.

Question 2

What is the null hypothesis in the context of the chi-squared test?

Answer 2

• The null hypothesis assumes that there is no statistically significant difference between sets of observations, meaning any observed difference is due to chance alone.

Question 3

In a coin toss example, what would the null hypothesis state?

Answer 3

• The null hypothesis would state that:
• there is no significant difference between the number of times the coin lands heads and the number of times it lands tails.

Question 4

What does the chi-squared test help determine in an experiment?

Answer 4

The chi-squared test helps determine whether any deviation between the observed and expected results is significant or if it can be attributed to chance.

Question 5

What are the four main criteria for using the chi-squared test?

Answer 5

1) The sample size must be large (over 20).
2) The data must fall into discrete categories.
3) Only raw counts, not percentages or rates, can be used.
4) It is used to compare experimental results with theoretical ones, such as Mendelian ratios in genetics.

Question 6

How do you calculate degrees of freedom in the chi-squared test?

Answer 6

• Degrees of freedom are calculated as the number of classes (categories) minus one.
• For example, if there are four blood types (A, B, AB, O), the degrees of freedom would be 3 (4 - 1).

Question 7

What is the formula for using chi-squared

Answer 7

Question 8

What role does the chi-squared distribution table play in the test?

Answer 8

The chi-squared distribution table is used to determine whether the calculated χ² value is significant by comparing it to values associated with different degrees of freedom and probabilities.

Question 9

In a chi-squared test, what does a p-value represent?

Answer 9

• The p-value represents the probability that the observed deviation from expected results is due to chance.
• It indicates the likelihood that the null hypothesis is true.

Question 10

What is the critical p-value in the chi-squared test, and what does it signify?

Answer 10

• The critical p-value is p = 0.05 (5%).
• If the probability of the deviation being due to chance is equal to or greater than 0.05, the null hypothesis is accepted.
• If the probability is less than 0.05, the null hypothesis is rejected, indicating the results are significant.

Question 11

If a calculated χ² value leads to a p-value greater than 0.05, what conclusion is drawn?

Answer 11

• If the p-value is greater than 0.05, it is assumed that the deviation is due to chance, and the null hypothesis is accepted, meaning the results are not significantly different from the expected.

Question 12

What conclusion would be drawn if the p-value was less than 0.05?

Answer 12

• If the p-value is less than 0.05, the deviation is considered significant, meaning that some factor other than chance is likely affecting the results.
• Therefore, the null hypothesis would be rejected.

Question 13

In the coin toss example, what conclusion would be reached if we observed 60 heads and 40 tails?

Answer 13

• A chi-squared test would result in a p-value slightly less than 0.05, leading to the rejection of the null hypothesis.
• This suggests that the coin might be biased or weighted.

Question 14

Why is the chi-squared test especially useful in genetics?

Answer 14

• The chi-squared test is useful in genetics to determine whether the observed genetic ratios, such as Mendelian inheritance patterns, deviate significantly from the expected ratios due to factors other than chance.

Question 15

In a genetic cross yielding a dihybrid F2 ratio, what would the null hypothesis state?

Answer 15

The null hypothesis would state that there is no significant difference between the observed genetic ratios and the expected 9:3:3:1 Mendelian ratio.

Question 16

How many degrees of freedom would you have in a genetic cross involving a 9:3:3:1 ratio, and why?

Answer 16

There would be three degrees of freedom because there are four classes (9:3:3:1), and degrees of freedom are calculated as the number of classes minus one (4 - 1 = 3).

Question 17

What would it mean if a χ² value in a genetic cross corresponds to a probability between 5% and 10%?

Answer 17

A probability between 5% and 10% indicates that the deviation from the expected results is not statistically significant, and the null hypothesis would be accepted. This means the variation could be due to chance.

Question 18

How would you interpret the result if the calculated χ² value falls between 6.25 and 7.82 for three degrees of freedom?

Answer 18

The value corresponds to a probability between 0.1 (10%) and 0.05 (5%), suggesting that the deviation is not statistically significant and the null hypothesis is accepted, meaning the results are not significantly different from the expected ratio.

Question 19

When conducting a chi-squared test, why is it important to know the number of degrees of freedom?

Answer 19

The number of degrees of freedom is crucial because it determines which row of the chi-squared distribution table to use when interpreting the significance of the calculated χ² value.

Question 20

What does it mean when a null hypothesis is accepted in a chi-squared test?

Answer 20

Accepting the null hypothesis means that there is no significant difference between the observed and expected results, and any deviation is likely due to chance.

Question 21

What does rejecting the null hypothesis indicate in a chi-squared test?

Answer 21

Rejecting the null hypothesis indicates that the deviation between observed and expected results is significant, suggesting that some factor other than chance is influencing the results.