Question

Which of the following statements regarding t and z distributions is/are true? Correctany false statements to make them true.

A) The area under a t-distribution to the left of -1.97 is greater than the area to the right of 2.17 for a sample of any size.

B) The area under the curve (AUC) to the right of t=2.00 when n=15 is smaller than the AUC to the right of t=2.00 when n=50.

C) The t-curve has thinner tails and a smaller standard deviation than a normal distribution for small sample sizes.

D) When x and (n-x) are both ≥ 5, the sampling distribution of p̂ is approximately normally distributed and we use a t coefficient to calculate the margin of error for estimating a sample proportion.

E) When we standardized the sampling distribution of sample means using estimatedSEM = s/√n, the result is distributed as a standard normal distribution when n=35.

Answer 1

Solution:-

A) True: The area under a t-distribution to the left of -1.97 is greater than the area to the right of 2.17 for a sample of any size.

B) False: The area under the curve (AUC) to the right of t = 2.00 when n = 15 is smaller than the AUC to the right of t = 2.00 when n = 50.

D.F = 14, t = 2.00

p-value = 0.033

D.F = 49, t = 2.00

p-value = 0.026

C) False: The t-curve has thinner tails and a smaller standard deviation than a normal distribution for small sample sizes.

The t-curve has thicker tails and a larger standard deviation than a normal distribution for small sample sizes.

D) False: When x and (n-x) are both ≥ 5, the sampling distribution of p̂ is approximately normally distributed and we use a t coefficient to calculate the margin of error for estimating a sample proportion.

When x and (n-x) are both ≥ 5, the sampling distribution of p̂ is approximately normally distributed and we use a z coefficient to calculate the margin of error for estimating a sample proportion.

E) True: When we standardized the sampling distribution of sample means using estimated SEM = s/√n, the result is distributed as a standard normal distribution when n=35.