A quality characteristic of interest for a teabagfilling pro

A quality characteristic of interest for a tea-bag-filling process is the weight of the tea in the individual bags. For company X, the label weight on the package indicates that the mean amount is 5.25 grams of tea in a bag. If the bags are under-filled, two problems arise. First, customers may not be able to brew the tea as strong as they wish. Second, the company may be in violation of the truth-in-labeling laws. On the other hand, if the mean amount of tea in a bag exceeds the label weight the company is giving away the product. Getting an exact amount of tea in a bag is difficult because of variation in the temperature and humidity inside the factory, differences in density of the tea, and extremely fast filling operation of the machine (about 180 bags per minute). The following table reports the weights in grams of 24 randomly selected tea bags produced by the company. (5.53,5.58,5.49,5.56,5.62,5.52,5.6,5.45,5.67,5.52,5.59,5.57,5.59,5.36,5.5,5.42,5.38,5.47,5.62,5.5,5.6,5.5,5.61,5.5.  Pls construct and interpret 90% and 99% confidence interval for the population mean weight of the tea bags, clearly showing all necessary steps.What would be the probability that the population mean weight of tea bags could be less than lower limit of 99% confidence interval? What do you think will happen to margins of error and confidence intervals if the sample size was to be increased to 35 observations? B. Is the company meeting the requirement set forth on label that the mean amount of tea in tea bags is 5.25 grams?

Solution

First let us find the sample mean and sample std deviation for the sample given.

Sample size n = 24

Mean = Average of X = 5.53125

Std dev of X = 0.07721

Std error of X = Std dev / rt 24=0.0158

As no of observations is only 24 < 30, we use critical t value to find out the confidence interval

df= 24-1 = 23

90% df 23 critical value of t = 2.0686

99% df 23 critical value of t = 3.1040

Margin of error = critical t * std error

Interpretation is we are 90% of 95% confidence that the sample mean lies within this confidence interval at 90 and 95% respectively

99% confidence level gives margin of error as 0.049043 which means sample will mean will differ from 5.25 by not more than 0.049043

If H0: mu = 5.25

   Ha: mu not equals 5.25

then t statistic = 17.80

p value = 0.00000 <0.00001

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If sample size increases, std error decreases and hence confidence interval becomes narrower

B) No The company is not meeting the requirement set forth on label that the mean amount of tea in tea bags is 5.25 grams