A random sample is obtained from a population with a variance of σ^2=400, and the sample mean is

step 1 Okay, so here we have that our null hypothesis is that mu is equal to 80, and then our alternati

A random sample is obtained from a population with a...

A random sample is obtained from a population with a variance of $\sigma^2=400$, and the sample mean is computed to be $\bar{x}_c=70$. Consider the null hypothesis $H_0: \mu=80$ versus the alternative hypothesis $H_1: \mu<80$. Compute the $p$-value for the following options. a. Sample size $n=25$ b. Sample size $n=16$ c. Sample size $n=44$ d. Sample size $n=32$

A random sample is obtained from a population with a variance of $\sigma^2=400$, and the sample mean is computed to be $\bar{x}_c=70$. Consider the null hypothesis $H_0: \mu=80$ versus the alternative hypothesis $H_1: \mu<80$. Compute the $p$-value for the following options.
a. Sample size $n=25$
b. Sample size $n=16$
c. Sample size $n=44$
d. Sample size $n=32$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide whether there is enough evidence in a sample to support a particular belief about a population parameter. It typically involves setting up a null hypothesis, which represents a default condition, and an alternative hypothesis, which represents a competing claim about the parameter.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) posits that there is no effect or that a specific parameter equals a certain value, while the alternative hypothesis (H?) suggests that the parameter differs in a certain direction. In this context, H?: ? = 80 and H?: ? < 80 indicate a one-sided test where the interest lies in detecting if the actual mean is less than 80.

Z-Test

A Z-test is used in hypothesis testing when the population variance is known and the sample is drawn from a normally distributed population, or when the sample size is large enough for the Central Limit Theorem to apply. It involves calculating a Z-statistic by standardizing the difference between the sample mean and the hypothesized mean.

Standard Error

The standard error is a measure of the variability of the sample mean around the population mean and is computed as the population standard deviation divided by the square root of the sample size. It quantifies how much the sample mean is expected to vary due to random sampling variability.

P-value

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic at least as extreme as the one that was actually observed. In a one-tailed test, it represents the area in the tail of the distribution beyond the calculated test statistic.

One-Tailed Test

A one-tailed test is used when the alternative hypothesis specifies a direction of the effect. In this case, since the alternative hypothesis H?: ? < 80 implies looking only for values lower than 80, the p-value is calculated using the left tail of the standard normal distribution.

Effect of Sample Size

The sample size affects the standard error, with larger samples leading to smaller standard errors. A smaller standard error increases the magnitude of the Z-statistic when the difference between the sample mean and the hypothesized mean remains the same, which in turn can result in a smaller p-value, indicating stronger evidence against the null hypothesis.

Transcript

Please give Ace some feedback