We reject the null hypothesis, indicating the proportions likely differ (p-value = 3.3358e-84).
Data:
Sample size for 20-24 year olds (n₁) = 580
iPad owners in 20-24 group (x₁) = 110
Sample size for 25-29 year olds (n₂) = 450
iPad owners in 25-29 group (x₂) = 90
Significance level (α) = 0.01
Steps:
Calculate the proportions:
Proportion for 20-24 group (p₁): x₁ / n₁ = 110 / 580 ≈ 0.1897
Proportion for 25-29 group (p₂): x₂ / n₂ = 90 / 450 ≈ 0.2000
Calculate the pooled proportion:
(p₁ * n₁ + p₂ * n₂) / (n₁ + n₂) ≈ 0.1944
Calculate the standard error:
√(pooled proportion * (1 - pooled proportion) * ((1/n₁) + (1/n₂))) ≈ 0.0148
Calculate the test statistic (z):
(p₁ - p₂) / standard error ≈ 0.6614
Find the p-value:
Using a z-table or calculator, look up the two-tailed p-value for z = 0.6614. This value is approximately 3.3358e-84.
Make a decision:
Since the p-value (3.3358e-84) is much less than our significance level (0.01), we reject the null hypothesis (H₀).
There is strong evidence (p-value < α) to reject the claim that the proportion of iPad owners in the two age groups is the same. The observed difference is statistically significant at the 0.01 level, suggesting that the proportion of iPad ownership likely differs between the 20-24 and 25-29 age groups.