It is given by the Central Limit Theorem that the increasing size of a population sample is accompanied by closer approximation of the distribution of the sample means to a normal distribution, regardless of the distribution of the "parent" population.
In general, the Central Limit Theorem holds for sample size n ≥ 30. If the population itself has a normal distribution, then the sampling distribution of the means is normal for any sample size.
With increasing sample size, the distribution of sample means converges toward the center of the distribution; the standard deviation of the sample means therefore decreases as the sample size increases.
The Central Limit Theorem shows that the standard deviation of the sample means σx-bar is equal to σ/√n where σ is the population's standard deviation and n is the sample size. The standard deviation of the sample means is called the standard error of the mean. The z-score for sample means is found by zx-bar equal to (x-bar − μ)/σx-bar where x-bar is the sample mean.
For this particular fruit with population mean μ of 772 grams and population standard deviation σ of 7 grams (with a sample size of 13), the standard deviation of the sample means σx-bar =7/√13.
Next, compute z768 equal to (768−772)/(7/√13) and z770 equal to (770−772)/(7/√13).
For z768 = -4√13/7 and z770 = -2√13/7, P(768 < x-bar < 770) equals P(-4√13/7 < zx-bar < -2√13/7).
A highly accurate program for Casio Programmable Calculators gives, for z768 = -4√13/7 and z770 = -2√13/7, corresponding proportions of area under the standard normal curve of -0.4803157823 and -0.3485319633; these areas are also the upper and lower boundaries of the probability sought with a difference of 0.131783819 equivalent to 0.1318.
