Please help me :(

Question

In a survey, 31 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $37.1 and standard deviation of $14.7. Estimate how much a typical parent would spend on their child's birthday gift (use a 90% confidence level). Give your answers to 3 decimal places. 

Express your answer in the format of ¯xx¯ ±±  Error. 

$ ±± $

Answer 1

Ok, let us start with the basics. We identify the known parameters.

Sample size = 31. Sample mean (X bar) = 37.1. Sample standard deviation s = $14.7

Definitions:

1. population - overall group being sampled
2. xbar - x bar or mean - average of the sample, which is also the expected value
3. or best guess of the population mean E(µ)
4. ∑sample standard deviation - estimate of Population standard deviation made from
5. the sample data which follows the following formula.

s² = ∑ (x - xbar)² /(n-1).

Since the question is about predicting the behavior of a single typical parent rather than predicting the average behavior of all parents (i.e. a population mean) we use standard deviation and not standard error in computing our confidence interval. (using a t distribution)

CI = µ ± t_(α/2) x s. where E(μ) = 37.1, s = 14.7; d.f. = n-1 or 30 ; t_{( .95,30)} = 1.697261

CI = [37.1 - 1.697261*14.7 , 37.1+ 1.697261*14.7]. or rounded to 3 decimal places

CI₉₀ = [$12.150 $62.050]

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Although not part of the answer here is how you would calculate the CI for mu (μ). Almost exactly the same but using the standard error or standard deviation of the mean where σ_xbar = s/√n or 14.7/5.568 (2.640)

This CIµ_{90 =} [ $32.619 $41.581]

This is not the range of prices a parent would pay but the range of the actual overall average for the entire population.