Q 8.17. - Elementary Statistics

Question

Wedding Costs. According to Bride's Magazine, getting married these days can be expensive when the costs of the reception, engagement ring, bridal gown, and pictures-just to name a few-are included. A simple random sample of $20$ recent U.S. weddings yielded the following data on wedding costs, in dollars. 

a. Use the data to obtain a point estimate for the population mean wedding cost. $\mu$, of all recent U.S. weddings. (Note: The sum of the data is $\$526,538$.)

b. Is your point estimate in part (a) likely to equal $\mu$exactly? Explain your answer.

Step-by-Step Solution

Answer

Part (a) $\$26326.9$

Part (b) No.

Part (a) Step 1: Given information

19,496	23,789	18312	14554	18460
27806	21203	29288	34081	27896
30098	13360	33178	42646	24053
32269	40406	35050	21083	19510

Part (a) Step 2: Concept

The formula used: Sample mean $=\frac{1}{n}\times \sum _{i=1}^{20}{x}_i$

Part (a) Step 3: Calculation

We've been handed a $20-$person random sample.

Let $x_i$ be the ${\tau }^{\text{th }}$ observation on wedding cost of the sample, for all $i=1,2,\dots ,20$

$\therefore$ Total wedding cost $=\sum _{i=1}^{20}{x}_i$

$=526538$ [Sum of the sample observation]

$\therefore$ Sample mean$=\frac{1}{n}\times \sum _{i=1}^{20}{x}_i,n=$ sample size $=20$

$=\frac{1}{20}\times 526538$

$=26326.9$

$\therefore$ Mean wedding cost for the given sample is $\$26326.9$

The sample mean can be used to estimate the (point) population mean wedding cost.

$\therefore$ The estimated value of the population mean $\left(\hat{\mu }\right)$

$=\$26326.9$

Part (b) Step 1: Explanation

No, it's doubtful that the sample mean $\overline{x}$ will be exactly the same as the population mean $\mu$ We're attempting to guess (estimate) the population mean value solely by drawing a sample, or a subset of the population. As a result, some sampling error is unavoidable.