Respuesta :
Answer:
Correct option: a. The null hypothesis is rejected.
Step-by-step explanation:
A statistical hypothesis test for difference between two means can be used to determine whether the mean time that families in Gotham have been living in their current homes is less than families in Metropolis.
The hypothesis is:
H₀: The mean time that families in Gotham in their current homes is not less than families in Metropolis,i.e. μ₁ ≥ μ₂.
Hₐ: The mean time that families in Gotham in their current homes is less than families in Metropolis,i.e. μ₁ < μ₂.
Given:
[tex]\bar X_{G}=35\\\bar X_{M}=50\\S_{G}^{2}=900\\S_{M}^{2}=105\\n_{G}=100\\n_{M}=150[/tex]
Assuming that the significance level of the test is, α = 0.10.
The test statistic is:
[tex]t=\frac{\bar X_{G}-\bar X_{M}}{s_{p}\sqrt{\frac{1}{n_{G}}+\frac{1}{n_{M}}}}[/tex]
Here s[tex]_{p}[/tex] = pooled standard deviation.
Compute the value of s[tex]_{p}[/tex] as follows:
[tex]s_{p} = \sqrt{\frac{(n_{G}-1)S_{G}^{2}+(n_{M}-1)S_{M}^{2}}{n_{G}+n_{M}-2}}=\sqrt{\frac{(100-1)900+(150-1)105}{100+150-2}}=20.55[/tex]
Compute the test statistic value as follows:
[tex]t=\frac{\bar X_{G}-\bar X_{M}}{s_{p}\sqrt{\frac{1}{n_{G}}+\frac{1}{n_{M}}}}=\frac{35-50}{20.55\sqrt{\frac{1}{100}+\frac{1}{150}}}=-5.654[/tex]
The test statistic value is -5.654.
The critical value of the test is, [tex]t_{\alpha, (n_{G}+n_{M}-2)}=t_{0.10,(100+150-2)}=t_{0.10,248}=-1.282[/tex]
*Use a t-table for the value.
*The negative sign is because of the test is left-tailed.
*Since the table does not consist the value for the degrees of freedom 248 use the next highest value.
Decision rule:
If the test statistic value is less than the critical value then the null hypothesis is rejected.
Test statistic = -5.654 < Critical t = -1.282.
Thus, the null hypothesis is rejected.
Using the t-distribution, it is found that the correct option is:
a. The null hypothesis is rejected.
At the null hypothesis, it is tested if the mean time that families in Gotham have been living in their current homes is less than families in Metropolis, that is:
[tex]H_0: \mu_G - \mu_H \geq 0[/tex]
At the alternative hypothesis, it is tested if it is less, that is:
[tex]H_0: \mu_G - \mu_H < 0[/tex]
The standard errors are:
[tex]s_G = \sqrt{\frac{900}{100}} = 3[/tex]
[tex]s_M = \sqrt{\frac{105}{150}} = 0.8367[/tex]
The distribution of the difference is given by:
[tex]\overline{x} = \mu_G - \mu_H = 35 - 50 = -15[/tex]
[tex]s = \sqrt{s_G^2 + s_M^2} = \sqrt{3^2 + 0.8367^2} = 3.1145[/tex]
The t-test statistic is given by:
[tex]t = \frac{\overline{x} - \mu}{s}[/tex]
In which [tex]\mu = 0[/tex] is the value tested at the null hypothesis.
Hence:
[tex]t = \frac{\overline{x} - \mu}{s}[/tex]
[tex]t = \frac{-15 - 0}{3.1145}[/tex]
[tex]t = -4.82[/tex]
The p-value is found using a left-tailed test, as we are testing if the mean is less than a value, with t = -4.82 and 150 + 100 - 2 = 248 df.
Using a t-distribution calculator, this p-value is of approximately 0, hence the null hypothesis is rejected and option a is correct.
A similar problem is given at https://brainly.com/question/14937586
