Final answer:
The distribution of sample means for size 5 is approximately normal due to the Central Limit Theorem, with the same mean as the population (2.5) and a standard deviation of about 0.863.
Step-by-step explanation:
The shape of the distribution of sample means for simple random samples of size 5 drawn from a geometric distribution with p = 0.4 would be approximately normal according to the Central Limit Theorem (CLT). The CLT states that the sampling distribution of the sample mean will be normally distributed, regardless of the shape of the population distribution, provided the sample size is sufficiently large (usually n > 30 is considered sufficient, but for non-severe non-normal distributions, smaller sample sizes can still approximate normality).
The mean of the sampling distribution will be the same as the population mean, which in this case is 2.5. The standard deviation of the sampling distribution, also known as the standard error (SE), can be calculated by dividing the population standard deviation by the square root of the sample size (n). Therefore, the standard error would be 1.93 / √5, which is approximately 0.863.
When comparing the means for different sample sizes, the theoretical mean would remain the same, but the standard deviation will decrease with larger sample sizes due to the formula SE = σ / √ n. The distribution of sample means will also become more closely approximated to a normal distribution as the sample size increases.