Math Problem Statement
på dansk
Solution
The image presents a statistical problem in Danish related to the germination rate of tulip bulbs. Here's a breakdown of the problem and how to solve it:
Problem Overview
A tulip nursery produces tulip bulbs, and it is assumed that 75% of the bulbs are capable of germination (spiringsdygtige). The bulbs are sold in bags containing 40 bulbs, randomly selected from the nursery’s production.
Part A
- Calculate the probability that at least 35 of the bulbs in a randomly selected bag are capable of germination, assuming the 75% germination rate.
Part B
- Calculate the mean and the standard deviation of the stochastic variable, which indicates the number of germinable bulbs in a bag.
Part C
- A customer does not believe that 75% of the bulbs are germinable. He buys a bag of bulbs, plants them, and finds that only 25 out of 40 bulbs germinate.
- Investigate whether the hypothesis that at least 75% of the bulbs are germinable (: At least 75% of the bulbs are germinable) can be rejected based on the customer’s experience, with a 5% significance level.
Solution Approach:
A. Probability Calculation
We can model this as a binomial distribution because each bulb either germinates or doesn't (a success or failure).
- The probability of success (germinable) is .
- The number of trials (bulbs in a bag) is .
The probability we seek is , where is the number of germinable bulbs, following a binomial distribution: We'll calculate by summing the binomial probabilities from to .
B. Mean and Standard Deviation
For a binomial distribution, the mean and standard deviation are calculated as follows: Substitute and to get the values.
C. Hypothesis Test
The null hypothesis is that at least 75% of the bulbs are germinable. The customer found that 25 out of 40 bulbs germinated, which is .
We will conduct a one-sample proportion test to determine whether this observed proportion is significantly lower than the assumed proportion of 75%.
Steps:
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Null Hypothesis:
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Alternative Hypothesis:
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Significance Level:
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Test Statistic (Z-score): where and .
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Compare the Z-score to the critical value for a 5% significance level to determine whether to reject the null hypothesis.
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Math Problem Analysis
Mathematical Concepts
Binomial Distribution
Hypothesis Testing
Standard Deviation
Mean
Formulas
Binomial distribution: P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Mean: μ = n * p
Standard Deviation: σ = √(n * p * (1 - p))
Z-test for proportion: Z = (p̂ - p0) / √(p0 * (1 - p0) / n)
Theorems
Central Limit Theorem
Binomial Theorem
Hypothesis Testing Theorem
Suitable Grade Level
High School to Early University (Grades 10-12 or University Level Statistics)
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