MATHEMATICS COLLEGE

A null hypothesis is that the mean nose lengths of men and women are the same. The alternative hypothesis is that men have a longer mean nose length than women. A statistical test is done and the p-value is 0.225. Which of the following is the most appropriate way to state the conclusion? a. The mean nose lengths of the populations of men and women are identical. b. There is not enough evidence to say that the populations of men and women have different mean nose lengths. c. Men have a greater mean nose length. d. The probability is 0.225 that men and women have the same mean nose length

Answers

Answer 1

Answer:

Answer:

b. There is not enough evidence to say that the populations of men and women have different mean nose lengths.

See explanation below.

Step-by-step explanation:

Develop the null and alternative hypotheses for this study?

We need to conduct a hypothesis in order to check if the means for the two groups are different (men have longer mean nose length than women), the system of hypothesis would be:

Null hypothesis: $\mu_(men) \leq \mu_(women)$

Alternative hypothesis: $\mu_(men) > \mu_(women)$

Assuming that we know the population deviations for each group, for this case is better apply a z test to compare means, and the statistic is given by:

$z=\frac{\bar X_(men)-\bar X_(women)}{\sqrt{(\sigma^2_(men))/(n_(men))+(\sigma^2_(women))/(n_(women))}}$ (1)

z-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.

Let's assume that the calculated statistic is $z_(calc)$

Since is a right tailed test test the p value would be:

$p_v =P(Z>z_(calc))=0.225$

And we know that the p value is 0.225. If we select a significance level for example 0.05 or 0.1 we see that $p_v >\alpha$

And on this case we have enough evidence to FAIl to reject the null hypothesis that the means are equal. So then the best conclusion would be:

b. There is not enough evidence to say that the populations of men and women have different mean nose lengths.

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A manufacturer knows that their items have a normally distributed length, with a mean of 13.1 inches, and standard deviation of 4.1 inches. If 25 items are chosen at random, what is the probability that their mean length is less than 11.1 inches

Answers

Answer:

0.73% probability that their mean length is less than 11.1 inches

Step-by-step explanation:

To solve this question, we have to understand the normal probability distribution and the central limit theorem.

Normal probability distribution:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = (X - \mu)/(\sigma)$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central limit theorem:

The Central Limit Theorem estabilishes that, for a random variable X, with mean $\mu$ and standard deviation $\sigma$ , a large sample size can be approximated to a normal distribution with mean $\mu$ and standard deviation, which is also called standard error $s = (\sigma)/(√(n))$