A rifle manufacturer is creating a new sniper rifle and is interested in testing the accuracy of its new automatic sighting mechanism. The gun is fired at various equidistant targets, and the vertical and horizontal distance from the shot to the desired target (the error) is measured each time the gun is fired. Consider the horizontal error (in centimeters) to be a random variable x; assume that x follows a normal distribution with an unknown population mean μ and a unknown standard deviation of σ. The value of x is positive when the shot is to the right of the target and negative when the shot is to the left of the target. The rifle manufacturer would like to conduct a hypothesis test about µ. The rifle manufacturer wants µ = 0, because this implies that, on average, the sighting mechanism is accurate.
A rifle manufacturer is creating a new sniper rifle and is interested in testing the accuracy of its new automatic sighting mechanism. The gun is fired at various equidistant targets, and the vertical and horizontal distance from the shot to the desired target (the error) is measured each time the gun is fired. Consider the horizontal error (in centimeters) to be a random variable x; assume that x follows a normal distribution with an unknown population mean μ and a unknown standard deviation of σ. The value of x is positive when the shot is to the right of the target and negative when the shot is to the left of the target. The rifle manufacturer would like to conduct a hypothesis test about µ. The rifle manufacturer wants µ = 0, because this implies that, on average, the sighting mechanism is accurate.
A rifle manufacturer is creating a new sniper rifle and is interested in testing the accuracy of its new automatic sighting mechanism. The gun is fired at various equidistant targets, and the vertical and horizontal distance from the shot to the desired target (the error) is measured each time the gun is fired. Consider the horizontal error (in centimeters) to be a random variable x; assume that x follows a normal distribution with an unknown population mean μ and a unknown standard deviation of σ. The value of x is positive when the shot is to the right of the target and negative when the shot is to the left of the target. The rifle manufacturer would like to conduct a hypothesis test about µ. The rifle manufacturer wants µ = 0, because this implies that, on average, the sighting mechanism is accurate.
A rifle manufacturer is creating a new sniper rifle and is interested in testing the accuracy of its new automatic sighting mechanism. The gun is fired at various equidistant targets, and the vertical and horizontal distance from the shot to the desired target (the error) is measured each time the gun is fired. Consider the horizontal error (in centimeters) to be a random variable x; assume that x follows a normal distribution with an unknown population mean μ and a unknown standard deviation of σ. The value of x is positive when the shot is to the right of the target and negative when the shot is to the left of the target.
The rifle manufacturer would like to conduct a hypothesis test about µ. The rifle manufacturer wants µ = 0, because this implies that, on average, the sighting mechanism is accurate.
To answer the questions that follow, download an Excel®® spreadsheet containing 10 observed values of the variable x by clicking on the following words in bold: Download Excel File.
Go through the following steps to conduct a hypothesis test about the population mean.
In the following sample Excel sheet, enter the appropriate formulas for conducting the hypothesis test about µ when σ is unknown.
Note: The function STDEV is used for Excel versions 2007 and earlier. For later versions, STDEV was replaced by STDEV.S. However, STDEV can still be used in later versions of Excel. Similarly, TDIST and NORMSDIST were replaced by T.DIST and NORM.S.DIST.
A
B
C
D
1
Horizontal Error, x
Hypothesis Test about a Population Mean:
2
0.0091
σ Unknown Case
3
-0.0036
4
0.0288
Sample Size
_______ (eg : COUNTA(A2-A11)
5
0.0193
Sample Mean
_________- (also formula)
6
0.0012
Sample Standard Deviation
___________(= stdev...)
7
0.0019
8
0.0052
Hypothesized Value
0
9
-0.0033
Standard Error
_________ (eg =d6/d4)
10
0.0072
Test Statistic t
__________ (eg: d5-d8/d2)
11
0.0101
Degrees of Freedom
= D4-1
12
13
p-value (Lower Tail)
______ (eg =1-d14)
14
p-value (Upper Tail)
=1-D13
15
p-value (Two Tail)
________ eg (d4/sqrt(d6)
16
Suppose the rifle manufacturer suspects that the sighting mechanism, on average, is shooting to the right of the target.
Use the values produced by the proper formulas in your Excel sheet to conduct the following hypothesis test at a significance level of α = 0.10:
H₀: µ ≤ 0
Ha�: µ > 0
The p-value is ______ than α = 0.10. Therefore, you _______ reject the null hypothesis and you ______ conclude the sighting mechanism, on average, is shooting to the right of the target.
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
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