(Solved): As he does on the first business day of each month, Ted has just opened up the suggestion box that L ...

As he does on the first business day of each month, Ted has just opened up the suggestion box that Luke placed in the plant to attract ideas from the employees. Ted finds a message from a production worker who is very concerned about the company’s racquetball racquet machine: Dear Thorndike management: "I am getting very worried about the racquetball racquet machine. Some of the older workers say it was here 15 years ago when they started, and I think it is just about worn out. I know new machines are expensive, so I thought I would collect some numbers that might help you out in making a decision to replace it. I know that the weight of the racquets coming off the machine is a big deal, since all of the racquets are weighed and then you decide which ones will be “Light,” “Regular,” and whatever you’ve decided to call the heavier model. Anyway, I’ve been doing some research. On my own time, honest. From time to time during the day, I take the 30 racquets that have just come from the machine and weigh them myself. You’ve told us that some racquets weigh more than others, so I can understand that one racquet might weigh 245 grams, the very next one might weigh 230 grams, and then the next one off the line might be all the way up to 250 grams. Like I said before, I think the machine is worn out. The first 30 racquets I weighed yesterday weighed an average of 236.5 grams, the next 30 weighed an average of 243.1 grams. I think the machine is going crazy, and you should either get it fixed or replace it. My friends would get pretty upset if they knew I wrote this suggestion, since buying a brand-new machine could mean that nobody gets a Christmas bonus this year. But I’m a young guy and I’d like for you to be successful so that I can have some job security and take care of my family. If you end up having a problem with the weights of your racquets, I might end up being out of a job. I’d like to find out if maybe I’m missing something here, and if the machine really is OK or not." Luke asks if the machine is still working the same as when the decision was made to offer racquets in three different weight categories. Ted replies that it seems to be. The average weight is still 240 grams, the standard deviation is still 10 grams, although he isn’t too sure whether the weights are still normally distributed. (a) How likely is it that a similar sample of racquets would be that far below the population mean? What’s the probability that x <= 236.5? (b) How likely is it that a similar sample of racquets would be that far above the population mean? What’s the probability that x >= 243.1?

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