What is the null hypothesis in the chi-square test for gender and age of starting smoking?

Gender and age of starting smoking are independent

Gender and age of starting smoking are associated

The mean age is the same for males and females

Males and females smoke equally often

How is the expected frequency of a cell calculated?

(row total x column total) / grand total

(row total + column total) / grand total

Observed frequency / number of cells

Calculated chi-square = 14.462 and the critical value at 5% with 3 d.f. is 7.815. What is the conclusion?

Reject H0; gender and age of starting smoking are associated

Fail to reject H0; gender and starting age are independent

Reject H0; no association exists

The test is inconclusive because the table is not 2 x 2

2074 Asoj · 5 marks

Chi-square: smokers, gender and starting age

Short answer

Chi-square test of independence gives χ² = 14.462 with 3 df. The critical value at 0.05 is 7.815. Since 14.462 > 7.815, reject H₀. There is significant association between gender and age at first smoking.

A random sample of smokers was obtained and each individual was classified both with respect to gender and with respect to the age at which he/she first started smoking.

Age	Male	Female
<16	25	10
16-17	24	32
18-20	28	17
>20	19	34
Carry out a test of hypotheses to decide whether there might be an association between gender and the age at which an individual first smokes?

Chi-square test of independence between gender and the age at which smoking was started (4 × 2 table, N = 189).

H0: the two attributes are independent (no association). H1: the two attributes are associated (dependent).
Level of significance: α = 0.05
Test statistic: χ² = Σ (O − E)²/E, where E = (row total × column total)/grand total.

Observed frequencies with totals:

                  Male  Female   Total
<16                 25      10      35
16-17               24      32      56
18-20               28      17      45
>20                 19      34      53
Total               96      93     189

Expected frequencies:

                  Male  Female   
<16              17.78   17.22
16-17            28.44   27.56
18-20            22.86   22.14
>20              26.92   26.08

Terms (O − E)²/E:

                  Male  Female   
<16              2.934   3.029
16-17            0.694   0.717
18-20            1.157   1.194
>20              2.330   2.406

χ² = Σ (O − E)²/E = 14.462

Degrees of freedom = (r − 1)(c − 1) = (4 − 1)(2 − 1) = 3. Critical value: χ²(0.05; 3) = 7.815. Reject H0 if χ² > 7.815.
Decision: calculated χ² > tabulated χ² (14.462 > 7.815, p-value = 0.0023), so reject H0.

Conclusion: there is a significant association between gender and the age at which a person first starts smoking.

✓ Answer checked by Claude Sonnet 5.5

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