A random sample of 49 measurements from one population had a...

A random sample of 49 measurements from one population had a sample mean of $10,$ with sample standard deviation $3 .$ An independent random sample of 64 measurements from a second population had a sample mean of $12,$ with sample standard deviation $4 .$ Test the claim that the population means are different. Use level of significance 0.01. (a) Check Requirements What distribution does the sample test statistic follow? Explain. (b) State the hypotheses. (c) Compute $\bar{x}_{1}-\bar{x}_{2}$ and the corresponding sample distribution value. (d) Estimate the $P$ -value of the sample test statistic. (e) Conclude the test. (f) Interpret the results.

A random sample of 49 measurements from one population had a sample mean of $10,$ with sample standard deviation $3 .$ An independent random sample of 64 measurements from a second population had a sample mean of $12,$ with sample standard deviation $4 .$ Test the claim that the population means are different. Use level of significance 0.01.
(a) Check Requirements What distribution does the sample test statistic follow? Explain.
(b) State the hypotheses.
(c) Compute $\bar{x}_{1}-\bar{x}_{2}$ and the corresponding sample distribution value.
(d) Estimate the $P$ -value of the sample test statistic.
(e) Conclude the test.
(f) Interpret the results.

Key Concepts

Level of Significance

The level of significance, denoted by ?, is a threshold for determining whether the p-value is small enough to reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting a true null hypothesis. Commonly used levels of significance include 0.05, 0.01, etc.

Two-Sample t-Test

This concept involves comparing the means of two independent populations using sample data. The two-sample t-test is used when the population variances are unknown and can be estimated using sample standard deviations. It helps determine if there is a statistically significant difference between the population means.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide whether a claim about a population parameter should be rejected based on sample data. It involves formulating a null hypothesis (typically stating no difference or no effect) and an alternative hypothesis (stating a difference or effect), then using a test statistic to assess the evidence against the null hypothesis.

Test Statistic

The test statistic is a value calculated from sample data that is compared against a theoretical distribution to decide whether to reject the null hypothesis. In the context of comparing two sample means, it is computed by taking the difference between the sample means and standardizing that difference, taking into account the estimated standard errors.

Degrees of Freedom

Degrees of freedom refer to the number of independent values that can vary in the calculation of a statistic. For a two-sample t-test, degrees of freedom are used to determine the appropriate t-distribution. When sample sizes are not equal or variances are unknown, an approximation (like the Welch-Satterthwaite equation) is commonly used to estimate the degrees of freedom.

P-Value

The p-value is the probability, under the assumption of the null hypothesis being true, of obtaining a test statistic at least as extreme as the one actually observed. It is used to determine the strength of evidence against the null hypothesis. A small p-value, typically less than the pre-determined level of significance, indicates that the observed data are unlikely under the null hypothesis.

Interpreting Results

Interpreting the results involves drawing conclusions from the test statistic and p-value. If the p-value is less than the level of significance, it provides sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis, implying a statistically significant difference between the population means. Otherwise, one fails to reject the null hypothesis, indicating insufficient evidence to support a difference.

Transcript

00:01 Question number nine.

00:03 Question e.

00:05 When testing, i claim about the difference of population means we use the student's t distribution or the normal distribution.

00:16 If the population standard deviations alpha 1 and alpha 2 are both known, then the sample test statics follows the normal distribution.

00:28 Else it follows the student's distribution.

00:33 Since alpha 1 and alpha 2 are pose unknown, the sample test statics follows the student's t distribution.

00:46 Question number b, then i'll hypothesis state that the population means are equal or that there is no difference between the population means.

00:59 So h node to mu1 equal new 2.

01:07 The alternative hypothesis state the opposite of the null hypothesis according to the claim different or not equal.

01:18 So h1 to new 1 not equal new 2.

01:24 So the result will be h node to mu1 equal new 2 and h1 to mu 1 not equal mu 2 stream number c given h node 2 mu 1 equal mu 2 h 1 2 new 1 not equal mu 2 x 1 equal 1 equal 10 s 1 equal 3 n 1 equal 49 x 4 2 2 equal 2 equal 12 s 2 equal 4 9 x 4 2 2 equal 4 into equal 64 and gamma equal 0 .01.

02:47 Determine the difference in sample means xx41 xbore 1 minus xpore 2 equal 10 minus 12 equal minus 2 determine the test statics t equal x b0 minus x bar 2 over square root s 1 squared over n1 plus s2 squared over n2 equal 10 minus 12 over square root 3 over over 49 plus 4 bar 2 over 64 this equal minus 3 .030 question number d h node to mu 1 equal mu 2 and h1 to mu 1 not equal mu 2 and x x1 1 equal 10 s 1 equal 3 n1 equal 49 x2 4 equal 12 is 2, 24, n2 equals 64, and 0, 0 ,0, 0 ,000...

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