A Box Of 14 Parts Contains 5 Defective Parts
A Box Of 14 Parts Contains 5 That Are Defective A Worker Picks Parts
A box containing 14 parts has 5 defective parts and 9 non-defective parts. A worker picks parts one at a time and attempts to install them. The assignment requires calculating probabilities related to the worker's process and identifying the appropriate statistical measures for describing the data based on the scale of measurement.
Assignment instructions:
Find the probability that the worker has to pick five parts to find one that is good. Round your answer to four decimal places. Additionally, determine the appropriate measure(s) to describe the average, shape, and spread of the data according to the scale of measurement (nominal, ordinal, interval/ratio).
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Paper For Above instruction
The scenario presents a classic probability problem involving a finite population, where one must determine the likelihood of specific outcomes when selecting parts from a box with known defective and non-defective components. Additionally, it involves understanding the appropriate statistical measures for summarizing data depending on its measurement scale.
Probability Calculation
Given that the box contains 14 parts, with 5 defective and 9 non-defective parts, the probability that the worker picks a defective part on any single draw is \(\frac{5}{14}\), and the probability of picking a non-defective part is \(\frac{9}{14}\).
The specific probability asked for is: what is the probability that the worker must pick five parts to find one good? In other words, the first four parts are defective, and the fifth is non-defective. Since the parts are picked sequentially without replacement, the probability mass function for the negative hypergeometric distribution applies.
The probability that the first four parts are defective and the fifth part is good is:
\[
P(\text{first 4 defective, 5th good}) = \left( \frac{5}{14} \times \frac{4}{13} \times \frac{3}{12} \times \frac{2}{11} \right) \times \left( \frac{9}{10} \right)
\]
Calculating step by step:
\[
\frac{5}{14} \times \frac{4}{13} \times \frac{3}{12} \times \frac{2}{11} = \frac{5 \times 4 \times 3 \times 2}{14 \times 13 \times 12 \times 11}
\]
\[
= \frac{120}{14 \times 13 \times 12 \times 11}
\]
\[
14 \times 13 = 182; \quad 182 \times 12 = 2184; \quad 2184 \times 11 = 24024
\]
So,
\[
\frac{120}{24024} = \frac{10}{2002}
\]
Now, multiply by the probability that the fifth selected part is good:
\[
\frac{9}{10}
\]
Therefore, the probability:
\[
P = \frac{10}{2002} \times \frac{9}{10} = \frac{10 \times 9}{2002 \times 10} = \frac{90}{20020} = \frac{9}{2002}
\]
Expressed as a decimal:
\[
P \approx \frac{9}{2002} \approx 0.0045
\]
Rounded to four decimal places:
\[
\boxed{0.0045}
\]
Thus, the probability that the worker picks four defective parts and then finds a good one on the fifth pick is approximately 0.0045.
Appropriate Measures of Central Tendency, Shape, and Spread
Understanding the nature of data requires analyzing the scale of measurement. The data in this context relates to discrete counts of defective versus non-defective parts, which informs the choice of statistical summaries.
- Nominal scale: Data categorized into labels without inherent order (e.g., defect types). Not applicable here.
- Ordinal scale: Data with a ranked order, but without meaningful intervals (e.g., quality ratings). Not applicable here.
- Interval/ratio scale: Data with meaningful numeric differences and ratios (e.g., number of defective parts, probabilities).
Given that the data involves counts and probabilities, the most appropriate measures include:
- Mean (average): Suitable for interval/ratio data to represent the central tendency, such as average number of defective parts or average probability.
- Median: Can represent the middle value in ordered data, useful if the data is skewed.
- Mode: Indicates the most frequent outcome, applicable in categorical contexts, but less relevant for continuous measures here.
- Shape: Described through skewness and kurtosis, providing insight into distribution symmetry and peakedness.
- Spread: Assessed with measures like variance, standard deviation, and range, which quantify variability around the mean. These are meaningful for ratio data.
In this case, since the data involves counts and probabilities related to discrete variables, the mean and variance are particularly suitable for summarizing the average behavior and variability. However, measures such as skewness and kurtosis can also describe the distribution shape if multiple observations or simulations are used.
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