Question

If the population standard deviation σ is unknown, and the sample size is small i.e.; n≤30, the confidence interval for the population mean µ is based on

Select one:

a. The binomial distribution

b. The normal distribution

c. The hypergeometric distribution

d. The t-distribution

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Answer 1

Answer:

a and b is the correct answer ............... ..............................

Answer 2

The confidence interval for the population mean µ, when the population standard deviation σ is unknown and the sample size is small (n ≤ 30), is based on the t-distribution.

Therefore, the correct answer is d. The t-distribution.

When the population standard deviation is unknown, we estimate it using the sample standard deviation s. However, the distribution of the sample means tends to follow the normal distribution only when the sample size is large (i.e., n > 30).

For small sample sizes, we use the t-distribution instead of the normal distribution to construct confidence intervals for the population mean. The t-distribution is a family of distributions that takes into account the additional uncertainty due to estimating the population standard deviation using the sample standard deviation.

The shape of the t-distribution is similar to that of the standard normal distribution, but it has heavier tails, meaning that it allows for more variability in the sample means. As the sample size increases, the t-distribution becomes more and more similar to the normal distribution.

The critical values of the t-distribution are obtained from the t-table, which takes into account the degrees of freedom (df = n-1) and the desired level of confidence. The formula for the confidence interval for the population mean using the t-distribution is:

x̄ ± tα/2,s/√n

where x̄ is the sample mean, s is the sample standard deviation, n is the sample size, tα/2 is the critical value from the t-distribution with df = n-1 and the desired level of confidence, and ± is the plus or minus sign indicating the interval around the sample mean.

Therefore , correct answer is d. The t-distribution.

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