Suppose that a farmer claims that the weights of his tomatoes are approximately Normally distributed with a mean of 100 grams and a standard deviation of 10 grams. To test this claim, you take a random sample of 50 tomatoes and classify them into 4 categories: 0–90 grams, 90–100 grams, 100–110 grams, and 110+ grams. How many tomatoes should you expect in the 90–100 gram category, assuming that the farmer’s claim is true

A z-table also known as the standard normal distribution table, helps us to know the percentage of values that are below (or to the left of the Distribution) a z-score in the standard normal distribution.

As it is given that the mean of the tomatoes is 100 grams, while the standard deviation is 10 grams. Also, it is given that the distribution is normally distributed. therefore, the probability that the tomatoes will be in the range of 90–100 gram category is,

[tex]P(90 < X < 100)=P(X < 100)-P(X < 90)[/tex]

[tex]=P(z < \dfrac{100-100}{10})-P(X < \dfrac{90-100}{10})\\\\=0.5 - 0.1587\\\\= 0.3413[/tex]

Thus, the percentage of tomatoes that will weigh between 90-100 grams is 34.13%.

Now, as the sample is of 50 tomatoes, therefore, the number of tomatoes should you expect in the 90–100 gram category, assuming that the farmer’s claim is true are

[tex]34.13\%\ \ of\ 50\\\\=\dfrac{34.13}{100} \times 50\\\\= 17.065 \approx 17[/tex]

Hence, the number of tomatoes that are expected in the 90–100 gram category is 17 in a sample of 50 tomatoes.

Learn more about Z-table:

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