A population has a mean of 300 and a standard deviation of 70. Suppose a sample of size 125 is selected and is used to estimate
. Use z-table. What is the probability that the sample mean will be within +/- 8 of the population mean (to 4 decimals)? (Round z value in intermediate calculations to 2 decimal places.) What is the probability that the sample mean will be within +/- 17 of the population mean (to 4 decimals)? (Round z value in intermediate calculations to 2 decimal places.)
1 answer:
Answer:
a: 0.8884
b: 0.9934
Step-by-step explanation:
We have µ = 300 and σ = 70. The sample size, n = 125.
For the sample to be within 8 units of the population mean, we would have sample values of 292 and 308, so we want to find:
P(292 < x < 308).
We need to find the z-scores that correspond to these values using the given data. See attached photo 1 for the calculation of these scores.
We have P(292 < x < 308) = 0.8884
Next we want the probability of the sample mean to be within 17 units of the population mean, so we want the values from 283 to 317. We want to find
P(283 < x < 317)
We need to find the z-scores that correspond to these values. See photo 2 for the calculation of these scores.
We have P(283 < x < 317) = 0.9934
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Answer:
Step-by-step explanation:
Hello!
Given the data for the variables:
Y: Selling price of a house on the shore of Tawas Bay
X₁: Number of bathrooms of a house on the shore of Tawas Bay.
X₂: Square feet of a house on the shore of Tawas Bay.
X₃: Number of bedrooms of a house on the shore of Tawas Bay.
The multiple regression model is Y= α + β₁X₁ + β₂X₂ + β₃X₃ + εi
a. Using software I've entered the raw data and estimated the regression coefficients:
^α= a= -5531.01
Represents the mean selling price of the houses when 0 bathrooms, 0 square feet and 0 bedrooms.
^β₁= b₁= -1386.21
Represents the modification of the mean selling price of the houses when the number of bathrooms increases in one unit and the square feet and number of bedrooms remain unchanged.
^β₂= b₂= 60.28
Represents the modification of the mean selling price of the houses when the square feet increase in one unit and the number of bathrooms and bedrooms remain unchanged.
^ β₃= b₃= 54797.08
Represents the modification of the mean selling price of the houses when the number of bedrooms increase in one unit and the number of bathrooms and square feet of the houses remain unchanged.
^Y= -5531.01 -1386.21X₁ + 60.28X₂ + 54797.08X₃
b)
R²= 0.55
R²Aj= 0.49
The coefficient of determination gives you an idea of how much of the variability of the dependent variable (Y) is due to the explanatory variables. Each time you add another explanatory variable to the regression the coefficient increases regarding of real contribution of the new variable. This could lead to thinking (wrongly) that the new variables are good to explain the dependent variable.
The adjusted coefficient of determination is a correction made to the raw coefficient of determination to have a more unbiased estimation of the effect the independent variables have over the dependent variable.
⇒ As you can see both coefficient are around 50%, which means that these explanatory variables
c)
The standard error estimate, this is the estimate of the population variance of the errors. In the ANOVA is represented by the Mean Square of the errors (MME)
Se²= MME= 3837640577.01
Se= 61948.6931
d) and f)
For the hypotheses tests for each slope the t- and p-values are:
α: 0.05
β₁: t= -0.06; p-value: 0.9528 ⇒ Do not reject H₀, the test is not significant.
β₂: t= 2.56; p-value: 0.0180 ⇒ Reject H₀, the test is significant.
β₃: t= 2.28; p-value: 0.0326 ⇒ Reject H₀, the test is significant.
e)
H₀: β₁= β₂= β₃
H₁: At least one βi is different from the others ∀ i=1, 2, 3
α: 0.05
F= 9.03
p-value: 0.0004
⇒ Reject H₀, the test is significant.
I hope it helps!