A sample of 50 data values has a mean of 2.314 and a standard deviation of 97687
ID: 3277493 • Letter: A
Question
A sample of 50 data values has a mean of 2.314 and a standard deviation of 9768776 and is slightly skewed to the right. Complete the following: a) Determine a 99% confidence interval for Y, the mean percentage of ash by weight. [You do not need to use the computer if you have the data mean and variance from your previous output.] Test the research hypothesis that the mean is less than 2.70, using a Type I error probability of 01. Be sure to provide the p-value, and use it to make your decision. (Show formally all the parts of the hypothesis test procedure.) We know from previous work that the data are somewhat skewed. Does this affect the accuracy of the conclusions in parts (a) and (b)? Hint: think about the sample size. b) c)Explanation / Answer
a. Assume that the data com efrom arandomized survey, and thus values are likely to be independent. The 50 data values are far less than 10% of all values. Therefore, randomization and 10% condition are reasonably met. Though the nearly normal condition is not met, as distribution of 50 data values are slightly skewed, it i snot enough to be a concern, because sample size is large (n>30).Use Student's t-model with n-1=50-149 degrees of freedom and find one-sample t tinterval for mean.
The 99% c.i for population mean, mu=xbar+-talpha/2, df=n-1 (s/sqrt n), where, xbar is sample mean, t is t critical at alpha/2 (alpha=0.01, alpha/2=0.005), n is sample size, s is sample standard deviation.
=2.314+-2.68(0.9769/sqrt 50)
=(1.944,2.684)
b. Hypotheses: H0:mu=2.70 (mean is no different from 2.70)
H1:mu<2.70 (mean is less than 2.70)
Assumptions for hypothesis test are same as that of confidence interval.
Significance level=0.01
Test statistic, t=(xbar-mu)/(s/sqrt n), where, mu is population mean.
=(2.314-2.70)/(0.9769/sqrt 50)
=-2.79
p value at 49 df is 0.004.
Rejection rule states that reject null hypothesis if p value is less than alpha=0.01. Here, 0.004 is less than 0.01, therefore, reject null hypothesis and conclude that mean is less than 2.70.
c. For very small sample size, n<15 the data should follow normal distribution pretty closely., for moderate sample size, 15<n<40, the t model works well, as long as data are unimodal and and reasonably symmetric. When sample size is greater than 40 or 50, the t method are safe to use unless data are extremely skewed. Thus, the accuracy is not affected as sample size is 50.