The average family spends $78 at the grocery store each week with a standard deviation of $25.
What percentage of families spend between $28 and $50? Explain in context what this represents.
You can play around with Z table stuff at
davidmlane.com/hyperstat/z_table.html
Well, it seems like these families are quite thrifty shoppers, spending a mere $78 a week at the grocery store. Now, let's dive into the numbers.
To find out what percentage of families spend between $28 and $50, we need to calculate the Z-scores for these values.
The Z-score formula is:
Z = (X - μ) / σ
Where:
X is the value we're interested in (in this case, $28 and $50),
μ is the mean ($78),
and σ is the standard deviation ($25).
For $28:
Z1 = ($28 - $78) / $25
Z1 ≈ -2.00
For $50:
Z2 = ($50 - $78) / $25
Z2 ≈ -1.20
Now we consult the Z-table or use a handy-dandy online Z-score calculator.
Approximately, the area between Z1 and Z2 (which represents the percentage of families spending between $28 and $50) is about 76.74%.
In plain English, this means that around 76.74% of families spend between $28 and $50 at the grocery store each week. So, it's safe to say that a considerable portion of these families knows how to stretch their grocery budget while keeping their pantry stocked. Kudos to them!
To find the percentage of families that spend between $28 and $50 at the grocery store, we need to use the concept of the z-score and the standard normal distribution.
First, we need to calculate the z-scores for both $28 and $50. The formula for z-score is:
z = (x - μ) / σ
where:
- x is the value we want to calculate the z-score for (in this case, $28 and $50)
- μ is the mean or average spending ($78)
- σ is the standard deviation ($25)
For $28:
z1 = (28 - 78) / 25 = -2
For $50:
z2 = (50 - 78) / 25 = -1.12
Next, we need to find the area under the standard normal distribution curve between these two z-scores. This area represents the percentage of families that spend between $28 and $50.
Using a standard normal table or a calculator, we can find that the area to the left of z1 (which represents the area below $28) is approximately 0.0228, and the area to the left of z2 (which represents the area below $50) is approximately 0.1314.
To find the area between z1 and z2 (representing the area between $28 and $50), we subtract the smaller area (0.0228) from the larger area (0.1314):
Area between z1 and z2 = 0.1314 - 0.0228 = 0.1086
This means that approximately 10.86% of families spend between $28 and $50 at the grocery store each week.
In context, this represents the percentage of families that fall within the specified spending range, which can be considered as a measure of affordability or budget allocation for groceries.
To find the percentage of families that spend between $28 and $50 at the grocery store each week, we can use the concept of standard deviation and z-scores.
Step 1: Calculate the z-scores for both $28 and $50 using the formula: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
For $28,
z = ($28 - $78) / $25 = -2
For $50,
z = ($50 - $78) / $25 = -1.12
Step 2: Look up the percentage associated with each z-score in the standard normal distribution table (also called a z-table).
The percentage corresponding to a z-score of -2 is approximately 2.28%.
The percentage corresponding to a z-score of -1.12 is approximately 13.50%.
Step 3: Calculate the percentage between $28 and $50 by subtracting the percentage of the lower value from the percentage of the higher value.
Percentage between $28 and $50 = 13.50% - 2.28% = 11.22%.
Therefore, approximately 11.22% of families spend between $28 and $50 at the grocery store each week.
In context, this represents the proportion of families whose grocery spending falls within the specified range. It provides an understanding of the distribution of spending habits among families.