Goodtaste Ltd., a coffee manufacturer, has recently been prosecuted for selling an underweight 100g jar of coffee. You have been asked to give assistance to the quality control manager who is investigating the problem. Jars are filled automatically and the filling machine can be preset to any desired weight. For the 100g jars of coffee a weight of 101g is set. There is no subsequent checking of the weight of individual jars , although samples are occasionally taken to check for quality. The standard deviation will depend to a certain extent on the mean weight, but for weights between 90g and 110g it is virtually constant at 1.5g. You have been asked to apply your knowledge of the normal distribution to the problem. In particular you have been told that prosecution only occurs when the product is underweight by more than 2%, so you need to find the probability that such a weight could happen by chance. (a) Assuming that the mean weight is 101g, what proportion of jars are: (i) Under 100g in weight? (ii) Under 98g in wieght? (iii) Under 97g in weight? (iv) Over 100g? (b) What should the mean weight be set to in order that the probability of a jar weighing less than 98g is less than 0.1%?
Goodtaste Ltd., a coffee manufacturer, has recently been prosecuted for selling an underweight 100g jar of coffee. You have been asked to give assistance to the quality control manager who is investigating the problem. Jars are filled automatically and the filling machine can be preset to any desired weight. For the 100g jars of coffee a weight of 101g is set. There is no subsequent checking of the weight of individual jars , although samples are occasionally taken to check for quality. The standard deviation will depend to a certain extent on the mean weight, but for weights between 90g and 110g it is virtually constant at 1.5g. You have been asked to apply your knowledge of the normal distribution to the problem. In particular you have been told that prosecution only occurs when the product is underweight by more than 2%, so you need to find the probability that such a weight could happen by chance. (a) Assuming that the mean weight is 101g, what proportion of jars are: (i) Under 100g in weight? (ii) Under 98g in wieght? (iii) Under 97g in weight? (iv) Over 100g? (b) What should the mean weight be set to in order that the probability of a jar weighing less than 98g is less than 0.1%?
Goodtaste Ltd., a coffee manufacturer, has recently been prosecuted for selling an underweight 100g jar of coffee. You have been asked to give assistance to the quality control manager who is investigating the problem. Jars are filled automatically and the filling machine can be preset to any desired weight. For the 100g jars of coffee a weight of 101g is set. There is no subsequent checking of the weight of individual jars , although samples are occasionally taken to check for quality. The standard deviation will depend to a certain extent on the mean weight, but for weights between 90g and 110g it is virtually constant at 1.5g. You have been asked to apply your knowledge of the normal distribution to the problem. In particular you have been told that prosecution only occurs when the product is underweight by more than 2%, so you need to find the probability that such a weight could happen by chance. (a) Assuming that the mean weight is 101g, what proportion of jars are: (i) Under 100g in weight? (ii) Under 98g in wieght? (iii) Under 97g in weight? (iv) Over 100g? (b) What should the mean weight be set to in order that the probability of a jar weighing less than 98g is less than 0.1%?
Goodtaste Ltd., a coffee manufacturer, has recently been prosecuted for selling an underweight 100g jar of coffee. You have been asked to give assistance to the quality control manager who is investigating the problem. Jars are filled automatically and the filling machine can be preset to any desired weight. For the 100g jars of coffee a weight of 101g is set. There is no subsequent checking of the weight of individual jars , although samples are occasionally taken to check for quality. The standard deviation will depend to a certain extent on the mean weight, but for weights between 90g and 110g it is virtually constant at 1.5g. You have been asked to apply your knowledge of the normal distribution to the problem. In particular you have been told that prosecution only occurs when the product is underweight by more than 2%, so you need to find the probability that such a weight could happen by chance. (a) Assuming that the mean weight is 101g, what proportion of jars are: (i) Under 100g in weight? (ii) Under 98g in wieght? (iii) Under 97g in weight? (iv) Over 100g? (b) What should the mean weight be set to in order that the probability of a jar weighing less than 98g is less than 0.1%? (c) Write a short note to the Quality Control Manager summarising your results.
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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