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Question
A random sample of 64 students has a mean score of 78 with a standard deviation of 12. Find the 95% confidence interval for the population mean.
Dezmos
95% CI: (75.06, 80.94)
Standard error = 12 / √64 = 1.5
Margin of error = 1.96 × 1.5 = 2.94
78 ± 2.94 → (75.06, 80.94)
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Question
In a survey of 250 students, 145 said they prefer online textbooks. Construct a 90% confidence interval for the true proportion.
Dezmos
90% CI ≈ (0.529, 0.631)
p̂ = 145/250 = 0.58
SE = √[0.58(0.42)/250]
Margin ≈ 0.051
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You
Why did you use 1.96?
Dezmos
Because a 95% confidence interval uses the critical z-value 1.96, which leaves 2.5% in each tail of the standard normal distribution.
You
explain that simpler
Dezmos
95% of the normal curve sits between −1.96 and +1.96, so we use 1.96 when calculating the interval.
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