If a family has 6 children, in how many ways could the parents have 3 boys and 3 girls as children?
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Answer:
Step-by-step explanation:
Hello!
X: length of human pregnancies from conception to birth.
X~N(μ;σ²)
μ= 264 day
σ= 16 day
If the variable of interest has a normal distribution, it's the sample mean, that it is also a variable on its own, has a normal distribution with parameters:
X[bar] ~N(μ;σ²/n)
When calculating a probability of a value of "X" happening it corresponds to use the standard normal: Z= (X[bar]-μ)/σ
When calculating the probability of the sample mean taking a given value, the variance is divided by the sample size. The standard normal distribution to use is Z= (X[bar]-μ)/(σ/√n)
a. You need to calculate the probability that the sample mean will be less than 260 for a random sample of 15 women.
P(X[bar]<260)= P(Z<(260-264)/(16/√15))= P(Z<-0.97)= 0.16602
b. P(X[bar]>b)= 0.05
You need to find the value of X[bar] that has above it 5% of the distribution and 95% below.
P(X[bar]≤b)= 0.95
P(Z≤(b-μ)/(σ/√n))= 0.95
The value of Z that accumulates 0.95 of probability is Z= 1.648
Now we reverse the standardization to reach the value of pregnancy length:
1.648= (b-264)/(16/√15)
1.648*(16/√15)= b-264
b= [1.648*(16/√15)]+264
b= 270.81 days
c. Now the sample taken is of 7 women and you need to calculate the probability of the sample mean of the length of pregnancy lies between 1800 and 1900 days.
Symbolically:
P(1800≤X[bar]≤1900) = P(X[bar]≤1900) - P(X[bar]≤1800)
P(Z≤(1900-264)/(16/√7)) - P(Z≤(1800-264)/(16/√7))
P(Z≤270.53) - P(Z≤253.99)= 1 - 1 = 0
d. P(X[bar]>270)= 0.1151
P(Z>(270-264)/(16/√n))= 0.1151
P(Z≤(270-264)/(16/√n))= 1 - 0.1151
P(Z≤6/(16/√n))= 0.8849
With the information of the cumulated probability you can reach the value of Z and clear the sample size needed:
P(Z≤1.200)= 0.8849
n= 10.24 ≅ 11 pregnant women.
I hope it helps!
Answer:
Step-by-step explanation:
Hello!
Given the variables:
X₁: Median Age
X₂: Median Income
b) Considering it from a logical point of view, income changes with age, for example, the more experienced the worker is you would think he would get a better paying job than a younger worker who is just starting. Then the dependent variable will be the median income and the independent variable will be the median age.
a) and c)
To see if there is a linear regression between the median income and median age you have to conduct a hypothesis test for the slope. If the slope is equal to zero, there is no linear regression between the two variables, if it is different to zero, there is a regression between the two of them:
H₀: β=0
H₁: β≠0
α: 0.05
The estimated regression equation for this regression ^Yi= 20.01 + 0.50X
The standard deviation for the slope is Sb= 0.11
And the p-value for the test is 0.0022
The p-value is less than the level of significance I choose, so the decision is to reject the null hypothesis. You can conclude that there is a linear regression between these two variables.
The correlation coefficient between the median income and the median age is r= 0.87 ⇒ This means you could expect a positive and strong linear correlation between the two variables.
d)
The slope represents how much the variable Y is modified whenever the variable X increases one unit.
In this example: Is the modification of the population mean of the median income, when the median age increases one year.
