Consider the following table:
Name | Length | Weight |
---|---|---|
Fritz | 187 | 85 |
Wilhelm | 161 | 66 |
Dieter | 163 | 66 |
Detlef | 195 | 98 |
What is the average of the variable “Length”?
The average “Length” is 176.5:
\[\bar x = \frac{187 + 161 + 163 + 195}{4} = 176.5.\]
A machine fills milk into \(250\)ml packages. It is suspected that the machine is not working correctly and that the amount of milk filled differs from the setpoint \(\mu_0 = 250\). A sample of \(157\) packages filled by the machine are collected. The sample mean \(\bar{y}\) is equal to \(240.8\) and the sample variance \(s^2_{n-1}\) is equal to \(137.67\).
Test the hypothesis that the amount filled corresponds on average to the setpoint. What is the value of the t-test statistic?
The t-test statistic is calculated by: \[ \begin{aligned} t & = & \frac{\bar y - \mu_0}{\sqrt{\frac{s^2_{n-1}}{n}}} = \frac{240.8 - 250}{\sqrt{\frac{137.67}{157}}} = -9.825. \end{aligned} \] The t-test statistic is thus equal to \(-9.825\).
In a small city the satisfaction with the local public transportation is evaluated. One question of interest is whether inhabitants of the city are more satisfied with public transportation compared to those living in the suburbs.
A survey with 250 respondents gave the following contingency table:
Location
Evaluation City Suburbs
Very good 18 22
Good 36 23
Bad 36 66
Very bad 10 39
The following table of percentages was constructed:
Location
Evaluation City Suburbs
Very good 18.0 14.7
Good 36.0 15.3
Bad 36.0 44.0
Very bad 10.0 26.0
Which of the following statements are correct?
In the percentage table, the column sums are about 100 (except for possible rounding errors). Hence, the table provides column percentages, i.e., conditional relative frequencies for satisfaction level given location type.
Consider the following regression results:
Call:
lm(formula = log(y) ~ x, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.45662 -0.31421 -0.04617 0.26287 1.27416
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.03675 0.06787 0.541 0.59
x -0.74279 0.05701 -13.030 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4987 on 52 degrees of freedom
Multiple R-squared: 0.7655, Adjusted R-squared: 0.761
F-statistic: 169.8 on 1 and 52 DF, p-value: < 2.2e-16
Describe how the response y
depends on the regressor x
.
The presented results describe a semi-logarithmic regression.
The mean of the response y
decreases with increasing x
.
If x
increases by 1 unit then a change of y
by about -52.42 percent can be expected.
Also, the effect of x
is significant at the 5 percent level.
For the 30 observations of the variable x
in the data file boxhist.csv draw a histogram, a boxplot and a stripchart. Based on the graphics, answer the following questions or check the correct statements, respectively. (Comment: The tolerance for numeric answers is \(\pm0.3\), the true/false statements are either about correct or clearly wrong.)