9.11
Test d'une affirmation concernant la moyenne : Écart type de la population connu
Une procédure complète pour tester une hypothèse concernant la moyenne d'une population est expliquée ici.
L'estimation de la moyenne d'une population…
L’exposition à différentes longueurs d’onde lumineuses peut affecter le taux de reproduction chez le poisson-zèbre.
Ainsi, une expérience est menée où un groupe de 50 poissons-zèbres est exposé à la lumière bleue, et leur taux de reproduction est comparé à celui du groupe témoin ayant la même taille d’échantillon.
Pour tester l’affirmation, nous commençons par l’hypothèse nulle selon laquelle le taux moyen de reproduction dans le groupe exposé et le groupe témoin est le même et une hypothèse alternative selon laquelle la lumière bleue augmente le taux de reproduction moyen.
L’expérience a montré que le taux de frai moyen dans le groupe exposé était de 550 par poisson, alors que pour le groupe témoin, il était de 250.
Le calcul de la statistique de test à partir de ces données nécessite une connaissance préalable de l’écart-type de la population, qui est de 146, connu des études précédentes.
À l’aide de ces données, nous pouvons calculer la statistique z et observer qu’elle se situe dans la région critique au niveau de signification de 0,05.
De plus, la valeur P de cette statistique z est inférieure à 0,05, ce qui conclut que la lumière bleue augmente le taux de frai chez le poisson zèbre.
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Q1: Why is population standard deviation important when testing a claim about the mean?
When population standard deviation is known, hypothesis testing about the mean becomes straightforward using the z distribution and normality assumption. Known standard deviation allows direct calculation of the z statistic from sample data, enabling comparison with critical values and significance levels. This method is rare in practice but highly efficient when the parameter is available from prior studies.
Q2: What conditions must be met before testing a hypothesis about a population mean?
Sample data must be collected from randomly selected samples with no sampling bias, and the sample size must exceed 30 to satisfy normality assumptions. The population standard deviation must be known beforehand. These conditions ensure the z distribution applies reliably and the test statistic accurately reflects the population parameter.
Q3: How do null and alternative hypotheses differ in a mean testing scenario?
The null hypothesis states that the population mean equals a specific value, representing no effect or change. The alternative hypothesis uses an inequality sign to claim the mean differs from that value. The direction of the inequality determines whether the test is right-tailed, left-tailed, or two-tailed, guiding the critical region placement.
Q4: What does it mean when a z statistic falls in the critical region?
When the calculated z statistic falls in the critical region at a chosen significance level like 0.05, it indicates the sample data provides strong evidence against the null hypothesis. This outcome leads to rejecting the null hypothesis and supporting the alternative hypothesis, suggesting a statistically significant effect exists.
Q5: How do the traditional method and P-value method compare in hypothesis testing?
The traditional method compares the calculated z statistic directly with the critical z score from the z table at a specified confidence level. The P-value method calculates the probability of observing sample data as extreme as or more extreme than what was observed, assuming the null hypothesis is true. Both methods lead to the same conclusion about rejecting or failing to reject the null hypothesis.
Q6: In the zebrafish spawning study, how did researchers conclude that blue light enhances spawning rate?
The z statistic calculated from the sample means (550 versus 250) and known population standard deviation of 146 fell within the critical region at the 0.05 significance level. Additionally, the P-value was less than 0.05, providing strong statistical evidence that blue light increases mean spawning rate compared to the control group.
Q7: When would you use testing a claim about mean with unknown population SD instead of known SD?
Most realistic situations involve unknown population standard deviation, requiring the t distribution instead of the z distribution. When population standard deviation is unavailable from prior research, you must estimate it from sample data. This approach is more common in practice than testing with known standard deviation.