How many solutions does the equation sin(5x) = 1/2 have on the interval (0, 2PI]
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Hello! You need to calculate a 99% confidence interval for the difference in mean lifespan between two tire brands. Each tested car was assigned one tire from each brand randomly on the rear wheels, allowing for paired sample analysis.
Brand 1 Brand 2 X₁-X₂
car 1: 36,925; 34,318; 2.607
car 2: 45,300; 42,280; 3.020
car 3: 36,240; 35,500; 0.740
car 4: 32,100; 31,950; 0.150
car 5: 37,210; 38,015; -0.0805
car 6: 48,360; 47,800; 1.160
car 7: 38,200; 37,810; 0.390
car 8: 33,500; 33,215; 0.285
n= 8
The study variable is defined as Xd= X₁-X₂, where X₁ represents the tire lifespan (in km) from Brand 1 and X₂ represents Brand 2. Thus, Xd is the difference in tire lifespan.
Xd~N(μd;δd²) (normality test p-value is 0.4640).
For calculating the confidence interval, the best statistic is the Student's t using the following formula:
t= (xd[bar] - μd)/(Sd/√n) ~t₍ₙ₋₁₎
sample mean: xd[bar]= 0.94
standard deviation: Sd= 1.29
= 3.355
xd[bar] ± 
*(Sd/√n) ⇒ 0.94 ± 3.355*(1.29/√8)
[-0.65;2.54]km.
The CI can be compared to bilateral hypothesis testing:
H₀:μd=0
H₁:μd≠0
using significance level of 0.01.
Since the confidence interval includes zero, we do not reject the null hypothesis, indicating no significant difference between the tire brands.
Hope you have a fantastic day!
Answer:
The variable x lies within the interval of all positive real numbers less than 5 cm.
Detailed solution:
Problem statement:
Determine the volume of the open-topped box as a function of the side length x (in centimeters) of the square cutouts.
Refer to the provided diagram for clarity.
Define:
x → length in centimeters of each square cutout side
The volume of the box with open top can be written as:
Given this, we have:
By substitution:
Determine the domain of x:
Because:
Therefore:
Domain is the interval (0,5)
That means all real numbers strictly greater than zero and less than 5 cm are valid for x.
Hence, the volume V as a function of x is:
