Summary data for two independent samples are p©û = 0.31, n©û = 400, and p©ü = 0.38, n©ü = 500. Use the 0.05 level of significance in testing H₀ : ¢³©û= ¢³©ü versus H©û : ¢³©û ¡Á ¢³©ü.
It would help, if you used the correct symbols. I don't know what you mean by ¢, ü or the copyright symbol.
Summary data for two independent samples are p1 = 0.31, N1 = 400, and p2 = 0.38, N2= 500. Use the 0.05 level of significance in testing Ho : ¥ð©û= ¥ð©ü versus H1 : ¥ð©û ¡Á ¥ð©ü.
Summary data for two independent samples are P1 = 0.31, N1= 400, and P2 = 0.38, N2= 500. Use the 0.05 level of significance in testing H1 : proportion1= proportion2 versus H1: porportion1 �‚ proportion2.
To test the hypothesis H₀ : p©û = p©ü versus H©û : p©û ≠ p©ü, we can use the two-sample z-test for proportions. The test statistic is calculated as:
z = (p©û - p©ü) / sqrt( p̂(1-p̂)(1/n©û + 1/n©ü) )
Where p̂ is the pooled sample proportion given by:
p̂ = (x©û + x©ü) / (n©û + n©ü)
Here, x©û and x©ü represent the number of successes in each sample, and n©û and n©ü represent the sample sizes.
Given the summary data:
p©û = 0.31, n©û = 400
p©ü = 0.38, n©ü = 500
We can calculate the pooled sample proportion:
p̂ = (0.31 * 400 + 0.38 * 500) / (400 + 500)
= 310 + 190 / 900
= 500 / 900
≈ 0.5556
Next, we can compute the test statistic:
z = (0.31 - 0.38) / sqrt(0.5556 * (1-0.5556) * (1/400 + 1/500))
= -0.07 / sqrt(0.3085 * 0.6915 * (0.0025 + 0.002))
≈ -0.07 / sqrt(0.1335 * 0.0045)
≈ -0.07 / sqrt(0.00060075)
≈ -0.07 / 0.024505
≈ -2.8555
The critical value for a two-tailed test at a 0.05 level of significance is approximately ±1.96. Since the test statistic (-2.8555) falls outside this range, we reject the null hypothesis.
Therefore, there is evidence to suggest that the proportions p©û and p©ü are significantly different at the 0.05 level of significance.