Please answer showing all calculations, & state the technique used if applicable
ID: 3291472 • Letter: P
Question
Please answer showing all calculations, & state the technique used if applicable. Thankyou.
Tests carried out on behalf of the Arborists Association on a random sample of twelve telescopic extension ladders manufactured by Safe GardenWorks yielded the following information.
X = 2,700 .0kgs x^2= 609,975.0 kgs^2
Here X is the maximum load that the ladder can support before it shows some sign of deformation.
(a) Find the 99% confidence limits of (i) the true mean and (ii) the true standard deviation of the probability distribution of X . Assume that X is a normally distributed random variable.
(b) Test the hypothesis that the mean value of the probability distribution of X is not significantly different from 220kgs,. Let = 0.05.
(c) Test the hypothesis that the true standard deviation of the probability distribution of X is not significantly different from 10kgs. Let = 0.05.
Explanation / Answer
Given that,
n = 12
X = 2,700 .0kgs x^2= 609,975.0 kgs^2
Here X is the maximum load that the ladder can support before it shows some sign of deformation.
First we have to find mean and standard deviation using these values.
Mean (Xbar) = X/n = 2700/12 = 225
variance = x^2 / n - mean2 = 609975.0/12 - 225^2 = 206.25
standard deviation =sqrt(variance) = sqrt(4827.27) = 14.36
(a) Find the 99% confidence limits of (i) the true mean and (ii) the true standard deviation of the probability distribution of X . Assume that X is a normally distributed random variable.
We can find 99% confidence interval for true mean by using TI-83 calculator.
steps :
STAT --> TESTS --> 8:TInterval --> ENTER --> Highlight on STats --> ENTER --> Input all the values --> C-level : 99.0 --> Calculate --> ENTER
99% confidence interval for true mean is (212.13,237.87).
Now we have to find 99% confidence interval for true standard deviation.
The 99% confidence interval for sigma is,
sqrt[(n-1)*s^2 / X2u ] < sigma < sqrt[(n-1)*s^2 / X2l ]
where X2u and X2l arecritical values for chi square distribution.
We can find critical values by using EXCEL.
syntax :
For X2u :
=CHIINV(probability,deg_freedom)
probability = (1 - C)/2
where C is confidence level = 99.0
deg_freedom = n-1 = 12-1 = 11
FOr X2l :
=CHIINV(probability,deg_freedom)
probability = (1 + C)/2
deg_freedom = 11
X2u = 26.76
X2l = 2.60
sqrt[(n-1)*s^2 / X2u ] < sigma < sqrt[(n-1)*s^2 / X2l ]
sqrt [(11*14.362)/26.76] < sigma < sqrt[(11*14.362) / 2.60]
9.21< sigma < 29.54
The 99% confidence interval for standard deviation is (9.21, 29.54)
(b) Test the hypothesis that the mean value of the probability distribution of X is not significantly different from 220kgs,. Let = 0.05.
Now we have to test the hypothesis that,
H0 : mu = 220 Vs H1 : mu not= 220
where mu is population mean.
Here we use one sample t-test.
We can do one sample t-test in TI-83 calculator.
steps :
STAT --> TESTS --> 2: TTest --> ENTER --> Highlight on Stats --> ENTER --> Input all the values --> Alternative : not equal --> Calculate --> ENTER
test statistic = 1.2062
P-value = 0.2531
P-value > alpha
Accept H0 at 5% level of significance.
Conclusion : There is not sufficient evidence to say that the mean value of the probability distribution of X is not significantly different from 220kgs.
c) Test the hypothesis that the true standard deviation of the probability distribution of X is not significantly different from 10kgs. Let = 0.05.
Here we have to test the hypothesis that,
H0 : sigma = 10 vs H1 : sigma not= 10
Here test statistic follows Chi-square distribution with n-1 dgerees of freedom.
The test statistic is,
X2 = (n-1) * s2 / sigma2
= (12-1)* / 14.362 / 102
= 22.69
Now we have to find p-value for taking decision.
P-value we can find in EXCEL.
syntax :
=CHIDIST(x, deg_freedom)
where x is test statistic
deg_freedom = n-1
P-value = 0.020
P-value < alpha
Reject H0 at 5% level of significance.
Conclusion : There is sufficient evidence to say that the true standard deviation of the probability distribution of X is not significantly different from 10kgs.