The height of a new species of plants have a mean of 2 m and a standard deviation of 40 cm. 100 samples of 50 plants each are measured. For how many samples would you expect the sample mean to be greater than 2.10 m?
I used normal distribution X~N(2, 400) since a sample have 50 plants so 0.4 m x 50 = 20, and the variance is 400. The distribution of sample mean is X~N(2, 4), but my answer is 0.4801 which means about 48 samples. The answer given is only 4 samples.
The mean for one sample is normally distributed with mean $2$ and standard deviation $\frac{0.4}{\sqrt{50}}=0.0565$. This means that $2.10$ metres is a Z-score of $\frac{0.1}{0.0565}=1.77$ and the probability the sample mean exceeds $2.10$ is $0.0384$. Since the $100$ samples are independent, the expected number of samples with mean exceeding $2.10$ is $0.0384×50=3.84$, which rounds to $4$.