Question: The mean amount of purchases by 25 typical customers at the Clinton grocery store is $23.50, with a standard deviation of 3.50. Assuming a normal distribution, the 90% interval is
In this essay, we will explore the 90% interval of purchases made by 25 typical customers at the Clinton grocery store. With the mean amount of purchases reported at $23.50 and a standard deviation of $3.50, we will assume a normal distribution to determine the range within which 90% of the purchases fall. Understanding this interval will provide valuable insights into the spending habits of customers at the store, aiding in decision-making and resource allocation for the business.
To establish the 90% interval, we rely on the properties of a normal distribution. As per the Central Limit Theorem, the distribution of the sample mean of sufficiently large samples (in this case, 25 customers) will approximate a normal distribution, regardless of the shape of the original population distribution. Given this, we can use the mean and standard deviation of the sample to estimate the mean and standard error of the population.
The standard error (SE) represents the standard deviation of the sample mean and can be calculated using the formula:
SE = standard deviation / sqrt(sample size)
In our case, the standard error (SE) is:
SE = 3.50 / sqrt(25) ≈ 0.70
Next, we need to find the critical Z-score that corresponds to the desired 90% confidence level. For a 90% confidence interval, the Z-score is approximately 1.645.
The margin of error (MOE) is calculated by multiplying the Z-score by the standard error:
MOE ≈ Z-score * SE
MOE ≈ 1.645 * 0.70 ≈ 1.155
Now, let’s apply the margin of error to the mean amount of purchases to determine the lower and upper bounds of the 90% interval.
Lower Bound ≈ Mean – MOE
Lower Bound ≈ $23.50 – $1.155 ≈ $22.345
Upper Bound ≈ Mean + MOE
Upper Bound ≈ $23.50 + $1.155 ≈ $24.655
Based on our analysis, the 90% interval for purchases by 25 typical customers at the Clinton grocery store is approximately between $22.35 and $24.66. Therefore, with 90% confidence, we can assert that most customer purchases lie within this range. Understanding this interval is crucial for the store’s management to make informed decisions about pricing strategies, inventory management, and promotional campaigns, ensuring customer satisfaction and maximizing profits.
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