Question

Garden. A company selling vegetable seeds in packets of 20 estimates that the mean number of seeds that will actually grow is 18 , with a standard deviation of \(1.2\) seeds. You buy 5 different seed packets. a) How many bad seeds do you expect to get? b) What's the standard deviation? c) What assumptions did you make about the seeds? Do you think that assumption is warranted? Explain.

Short answer

a) 10 bad seeds. b) Standard deviation is approximately 2.68. c) Assumed packets are independent, which is reasonable if conditions are consistent.

Step-by-step solution

Identify Mean and Standard Deviation for One Packet

The problem states that the mean number of seeds that will grow per packet is 18, and the standard deviation is 1.2 seeds. Each packet contains 20 seeds.

Determine the Expected Number of Bad Seeds for One Packet

For one packet, if 18 seeds are expected to grow, the number of bad seeds would be the remaining seeds in the packet. Since each packet has 20 seeds, the number of bad seeds per packet is \(20 - 18 = 2\) seeds.

Calculate Expected Number of Bad Seeds for 5 Packets

For 5 packets, multiply the expected number of bad seeds per packet by the number of packets: \(5 \times 2 = 10\) bad seeds.

Find the Combined Standard Deviation for 5 Packets

The standard deviation for multiple independent events (like packets) is the square root of the sum of the variances. For one packet, the variance is \( (1.2)^2 = 1.44 \). For 5 packets: \( \sqrt{5 \times 1.44} = \sqrt{7.2} \approx 2.68 \).

Discuss Assumptions About the Seeds

We assumed that the seed packets are independent, meaning the behavior of seeds in one packet does not affect others. This assumption is reasonable if the seeds are consistently packed under the same conditions without variation between packets.

Key concepts

Standard Deviation

Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. It indicates how much individual data points deviate from the overall mean, and is particularly useful in determining the reliability and predictability of data.
In our seed example, a standard deviation of 1.2 suggests that the number of seeds that grow is generally close to the average of 18, but there can be some fluctuation either above or below.
To calculate the standard deviation for multiple packets, we look at the variance first, which measures the average degree to which each number is different from the mean.
Variance per packet is calculated by squaring the standard deviation, showing the average squared differences from the mean. In this scenario, it is \[(1.2)^2 = 1.44.\]
For multiple independent packets, you add up the variances and then take the square root to find the overall standard deviation, giving you a measure on these combined events. In this problem with 5 packets, it gives\[\sqrt{5 \times 1.44} = \sqrt{7.2} \approx 2.68.\]This shows the variability of grown seeds across all packets.

Mean Calculation

The mean is a measure of central tendency, often referred to as the average. It is calculated by adding up all data points and then dividing by the number of points. In simple terms, it tells us where the center of a data set is located.
In our case, the mean number of seeds per packet that germinate is given as 18. Since each packet contains 20 seeds, we expect 2 seeds per packet not to grow, or remain as 'bad seeds'.
For multiple packets, we multiply the mean number of bad seeds per packet by the total number of packets to find the total expected number of bad seeds. Here,\(5 \times 2 = 10\) bad seeds for 5 packets.
This calculation assumes that the average number of seeds that fail to germinate stays constant across all packets, reflecting consistency.

Independent Events

Independent events are those whose outcomes do not affect one another. In statistics, this is an important concept because it allows us to simplify calculations and make predictions based on probabilities.
In our exercise, each seed packet is considered an independent event, meaning the number of seeds that germinate in one packet does not influence how many germinate in another.
This independence allows us to apply specific rules to calculate the expected total variance and standard deviation for the combined seed packets, as we did using the square root of the summed variances across the packets.
Assuming independence in this scenario is reasonable if all packets are produced under the same conditions, with no extraneous factors affecting specific packets differently. However, any inconsistency in conditions (like varying soil quality or storage conditions) might compromise the independence assumption, altering the predictability of outcomes.