Respuesta :
Answer:
Since our calculated value is lower than our critical value,[tex]z_{calc}=2.02<2.33=z_{critical}[/tex], we have enough evidence to FAIL to reject the null hypothesis at 1% of singificance. So there is not nough evidence to conclude that mean for men it's significantly higher than the mean for female.
Step-by-step explanation:
1) Data given and notation
[tex]\bar X_{1}=5.2[/tex] represent the mean for group men
[tex]\bar X_{2}=4.5[/tex] represent the mean for group women
Assuming these values for the remaining data:
[tex]\sigma_{1}=1.2[/tex] represent the population standard deviation for the sample men
[tex]\sigma_{2}=1.5[/tex] represent the population standard deviation for the sample women
[tex]n_{1}=32[/tex] sample size for the group men
[tex]n_{2}=30[/tex] sample size for the group women
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value
2) Concepts and formulas to use
We need to conduct a hypothesis in order to check if the mean for men it's higher than the mean for women, the system of hypothesis would be:
H0:[tex]\mu_{1} \leq \mu_{2}[/tex]
H1:[tex]\mu_{1} > \mu_{2}[/tex]
If we analyze the size for the samples both are higher than 30, and we know the population deviation's, so for this case is better apply a z test to compare means, and the statistic is given by:
[tex]z=\frac{\bar X_{1}-\bar X_{2}}{\sqrt{\frac{\sigma^2_{1}}{n_{1}}+\frac{\sigma^2_{2}}{n_{2}}}}[/tex] (1)
z-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.
3) Calculate the statistic
We have all in order to replace in formula (1) like this:
[tex]z=\frac{5.2-4.5}{\sqrt{\frac{1.2^2}{32}+\frac{1.5^2}{30}}}=2.02[/tex]
4) Find the critical value
In order to find the critical value we need to take in count that we are conducting a right tailed test, so we are looking on the normal standard distribution a value that accumulats 0.01 of the area on the right and 0.99 of the area on the left. We can us excel or a table to find it, for example the code in Excel is:
"=NORM.INV(1-0.01,0,1)", and we got [tex]z_{critical}=2.33[/tex]
5) Statistical decision
Since our calculated value is lower than our critical value,[tex]z_{calc}=2.02<2.33=z_{critical}[/tex], we have enough evidence to FAIL to reject the null hypothesis at 1% of significance. So there is not nough evidence to conclude that mean for men it's significantly higher than the mean for female.
There is not enough evidence to conclude that the mean for men it's significantly higher than the mean for females.
What are null hypotheses and alternative hypotheses?
In null hypotheses, there is no relationship between the two phenomenons under the assumption or it is not associated with the group. And in alternative hypotheses, there is a relationship between the two chosen unknowns.
Men are believed to have a slightly longer average length of short hospital stays than women; 5.2 days versus 4.5 days respectively.
[tex]\sigma _1 = 1.2\\[/tex], the population for sample men
[tex]\sigma _2= 1.5\\[/tex], the population for sample women
[tex]n_1 = 32[/tex], the sample size for men
[tex]n_2 = 30[/tex], the sample size for women
The mean for men is higher than the mean for women. Then
[tex]H_o : \mu _1 \leq \mu _2\\\\H_a : \mu _1 > \mu _2[/tex]
The size for the samples both are higher than 30, and we know the population deviations, so for this case is better to apply a z-test to compare mean, and the statistic is given as
[tex]z = \dfrac{\bar{X} _1 - \bar{X} _2}{\sqrt{\frac{\sigma _1^2}{n_1}+\frac{\sigma _2^2}{n_2}}}[/tex]
Then we have
[tex]z = \dfrac{5.5 - 4.5}{\sqrt{\frac{1.2^2}{32}+\frac{1.5^2}{30}}} = 2.02[/tex]
The critical value for z = 2.02 is 2.33
[tex]z _{calc} = 2.02 < 2.33 = z _{critical}[/tex], we have enough evidence to Fail to reject the null hypothesis at 1% of significance.
So there is not enough evidence to conclude that the mean for men it's significantly higher than the mean for females.
More about the null hypotheses and alternative hypotheses link is given below.
https://brainly.com/question/9504281
