In a survey, 20 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $40.3 and standard deviation of $5.8. Estimate how much a typical parent would spend on their child's birthday gift (use a 95% confidence level). Give your answers to 3 decimal places.Express your answer in the format of ¯xx¯ ±±  E. $ ±± $

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Asked Nov 23, 2019
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In a survey, 20 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $40.3 and standard deviation of $5.8. Estimate how much a typical parent would spend on their child's birthday gift (use a 95% confidence level). Give your answers to 3 decimal places.

Express your answer in the format of ¯xx¯ ±±  E. 
$ ±± $

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Expert Answer

Step 1

Introduction

The 100 (1 – α) % confidence interval for the population mean, μ, for given sample standard deviation, s is: ( – (tα/2; n – 1­) (s/√n),  + (tα/2; n – 1) (s/√n)).

Here, n is the sample size,  is the sample mean, and tα/2; n – 1 is the critical value of the t-distribution with (n – 1) degrees of freedom, above which, 100 (α/2) % or α/2 proportion of the observations lie.

The t-distribution is used, because the population standard deviation is unknown and the sample standard deviation is being used as a substitute.

Step 2

Calculation:

Here, n = 20;  = 40.3; s = 5.8.

Therefore, degrees of freedom = 20 – 1 = 19.

Again, 100 (1 – α) % = 95% = 0.95.

Thus, α = 0.05.

From the Excel formula: =T.INV.2T(0.05,19), /2; n – 1­ = t0.025; 19 = 2.093.

Thus, the confidence interval is calculated as follows:

( – (/2...

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