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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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