Given a sample of size π1=40 from a population with a standard deviation π1=20, and an independent sample of size π2= 50 from another population with a known standard deviation π2= 10, if the manager wants to test that the two population means differ, given π₯1= 72 and π₯2=66, which one of the following statements is not true:
1. The standard error is 3.4641.
2. The test statistic is 1.8532.
3. The rejection region at 5% level of significance is π§ <-1.96 and π§ > 1.96.
4. The statistical conclusion is we fail to reject HO.
5. HO: π1- π2=0 vs HO: π1- π2β 0Β
Population 1
"n_1=40\\\\\\bar x_1=72\\\\\\sigma_1=20"
Population 2
"n_2=50\\\\\\bar x_2=66\\\\\\sigma_2=10"
The hypotheses to be tested are,
"H_:\\mu_1-\\mu_2=0\\\\vs\\\\H_1:\\mu_1-\\mu_2\\not=0"
Since the population variances are known and the sample sizes for both populations are large "(\\gt 30)", we shall apply the standard normal distribution to perform this hypothesis test as follows.
The test statistic is given as,
"Z_c={(\\bar x_1-\\bar x_2)\\over SE(\\bar x_1-\\bar x_2)}" where "SE(\\bar x_1-\\bar x _2)" is the standard error of difference of the means given as,"SE(\\bar x_1-\\bar x_2)=\\sqrt{{\\sigma^2_1\\over n_1}+{\\sigma^2_2\\over n_2}}=\\sqrt{{400\\over40}+{100\\over50}}=\\sqrt{10+2}=\\sqrt{12}=3.4641(4dp)"
The test statistic is therefore,
"Z_c={72-66\\over3.4641}={6\\over 3.4641}=1.7321(4dp)"
The critical value is the standard normal table value at "\\alpha=0.05" given as,
"Z_{0.05\\over 2}=Z_{0.025}=1.96"
Since we are performing a two-sided test, we shall have two critical values namely, "-1.96" and "1.96." Therefore, the rejection region is, "Z\\lt-1.96" and "Z\\gt1.96".
The null hypothesis is rejected if, "|Z_c|\\gt Z_{0.025}"
For this case, "|Z_c|=1.7321\\lt Z_{0.025}=1.96", we fail to reject the null hypothesis and we conclude that there is no sufficient evidence to show that the means differ at 5% level of significance.
From the choices given above, the choice number (2) is not true.
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