Question

Sammy's Salsa, a small local company, produces 20 cases of salsa a day. Each case contains 12 jars and is imprinted with a code indicating the date and batch number. To help maintain consistency, at the end of each day, Sammy selects three jars of salsa, weighs the contents, and tastes the product. Help Sammy select the sample jars. Today's cases are coded 07N61 through $07\mathrm{N}80.$ a) Carefully explain your sampling strategy. b) Show how to use random numbers to pick 3 jars. c) Did you use a simple random sample? Explain.

Short answer

Use simple random sampling to select 3 jars by generating random numbers from 1 to 240. Map these numbers to specific jars in the batches.

Step-by-step solution

Understand the Codes

Today's cases are coded from 07N61 through 07N80. This means there are 20 cases (each case getting a consecutive number from 61 to 80). Each case contains 12 jars, so in total, we have 240 jars (20 cases * 12 jars per case).

Define the Sampling Strategy

To randomly select jars, we will use a simple random sampling method. This involves giving each jar an equal chance of being selected without replacement. Since each case is essentially another batch, selecting jars from dispersed cases may offer insights into batch consistency as well.

Generate Random Numbers

To select 3 jars randomly, we generate random numbers to represent jar numbers from 1 to 240. For example, using a random number generator, the numbers 45, 127, and 209 might come up. These numbers correspond to specific jars, where jar 45 could be in case 64 (as cases are sequential batches, each with an equal number of jars).

Map Random Numbers to Jars

The random numbers need to be converted to specific case and jar numbers. For instance, random number 45 corresponds to the 9th jar of the case 07N64 (since case 07N61 contains jars 1-12, case 07N62 contains jars 13-24, etc.). Follow the same process for all random numbers.

Confirm the Sampling Type

Yes, we used a simple random sampling technique because each jar had an equal probability of selection, and selection was made without replacement from the pool of available jars.

Key concepts

Random Number Generation

When selecting samples randomly, generating random numbers is a crucial step. Sammy's Salsa aims to pick 3 jars from 240 available jars, so random numbers must cover this range. Sammy can use a random number generator tool, either a physical device or online software, to produce these numbers. This tool will ensure that each number, hence each jar, has an equal chance of being chosen, ranging from 1 to 240.

The key here is to produce numbers that are truly random. True randomness means there isn't any bias, and each number is independent of the last one generated. This process makes sure each jar is considered equitably in the selection.

Sampling Strategy

The sampling strategy employed by Sammy's Salsa is focused on achieving a fair representation and maintaining quality control. A simple random sampling method was chosen because it gives each jar an equal opportunity to be selected for quality checks.

In simple random sampling, selection happens without replacement, meaning once a jar is picked, it won't be considered for that day's sample again. This is crucial to ensuring that there is a true reflection of the overall production and the quality check isn't biased by repeated jar sampling. This approach provides a broad overview of the batch, allowing Sammy to assess consistency across different cases effectively.

Using this strategy helps ensure that the sample is unbiased and gives Sammy actionable insights about the daily production quality.

Batch Consistency

Batch consistency involves evaluating the uniformity of a product across a specified batch of production. For Sammy, consistency is measured by randomly selecting and assessing jars from various cases, checking their weight and taste.

By sampling jars from different cases - effectively treating each case as a batch - Sammy can determine if there are variations in quality. If all sampled jars match the expected quality standards, it's likely that the batch is consistent.

This method helps in detecting anomalies or deviations early, ensuring that each batch meets the company’s standards and expectations. Monitoring batch consistency can greatly enhance customer satisfaction, ensuring that quality stays uniform across all salsa jars.

Random Selection Process

The random selection process involves several steps to ensure fairness and validity. After generating random numbers, each number has to be mapped to a specific jar. For Sammy's Salsa, jars are numbered sequentially across cases, from 1 to 240.

The mapping starts by identifying which case a random number belongs to. Each case holds 12 jars, so simple division or counting helps in determining the case number and the specific jar within the case. For example, random number 45 would be in the 4th case, 9th jar, following the sequential numbering system.

This process ensures that the selection is neither biased nor repetitive, thereby aligning with simple random sampling principles. Random selection processes like these help businesses evaluate products effectively without bias or redundant sampling, ensuring all products are considered equally.