Now suppose a new sample of university students were asked the question, “In hours, what was your phone screen time yesterday?” Some researchers wish to know if the average daily phone screen time of university students is significantly different from 4 hours. That is, they wish to test the hypotheses where denotes the mean average daily phone screen time of university students in hours. Using the data collected, a one-sample -test was carried out and the results were stored. Below are some facts about this test that will be useful for answering this question: = 78 df = 77 = 3.83 = 1.991 = 1.37 Using the above facts, your task is to calculate a 95% confidence interval. One of the options below is the correct 95% confidence interval. Choose the best answer, noting that there may be some minor differences between your answer and the best option below due to rounding:

Question

BlackTom AI · Accepted Answer

The 95% confidence interval for a population mean (when sigma is unknown) is x̄ ± t* (s / √n). Here x̄ = 3.83, s = 1.37, n = 78, and t* = t0.975,77 = 1.991.

Step-by-step calculation:
1. Compute the standard error (SE): SE = s / √n.
   √n = √78 ≈ 8.831760866.
   SE = 1.37 / 8.831760866 ≈ 0.155119.

2. Compute the margin of error (ME): ME = t* × SE.
   ME = 1.991 × 0.155119 ≈ 0.308842 (intermediate: 1.991×0.155119 = 0.310238 − 0.001396 ≈ 0.308842).

3. Compute the confidence limits:
   Lower limit = x̄ − ME = 3.83 − 0.308842 ≈ 3.52116.
   Upper limit = x̄ + ME = 3.83 + 0.308842 ≈ 4.13884.

Rounded to three decimal places, the 95% CI is approximately (3.521, 4.139).

Why this is correct:
- The t-distribution is used because the population standard deviation is unknown and sample size is moderate; the correct critical value t0.975 with df=77 is given as 1.991.
- SE properly accounts for sample size via √n; using s alone or an incorrect divisor would produce wrong widths.

Why the other options are incorrect:
- Option b (3.675, 3.985) has a much smaller margin (≈0.155), which corresponds to using SE but with t*≈1.0 rather than 1.991 (or an arithmetic mistake); it underestimates uncertainty.
- Option c (2.83, 4.83) has a margin of ±1.00, which is roughly s (1.37) scaled incorrectly (likely s used without √n); it overstates uncertainty.
- Option d (2.46, 5.2) is even wider and not consistent with the computed SE and given t*.

Conclusion: The computed 95% confidence interval (3.521, 4.139) matches option a.

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