Question

time left 0:45:47 question 2 not yet answered marked out of 3.00 flag question question texttitanium alloys have very high hardness. adding small amounts of copper tends to improve their properties, such as hardness. in order to evaluate the effect of copper addition on hardness of titanium alloys, 10 random casts with 5% copper were made in an argon-arc melting furnace. the rockwell hardness was determined for each of the casts. then, the casts were subjected to high temperature (ht) and hardness was determined again in the second row. assume hardness follows a normal distribution. use the data below to answer the next three questions. hard5 248 247 245 247 248 250 247 246 243 244 hard5ht 250 247 246 251 251 253 248 246 249 247 (a) do the data provide evidence that high temperature increases the hardness of 5% copper alloys, on average? what is the distribution of the test statistic under the null hypothesis and the exact p-value? t-distribution with df

Answer 1

The question involves using a t-test to analyze whether high temperature increases hardness in a titanium-copper alloy. The test statistic follows a t-distribution, and one would calculate the p-value based on this distribution.

Step-by-step explanation:

The question asks whether the data provide evidence that high temperature increases the hardness of 5% copper-titanium alloys on average. To evaluate this, we use a t-test to compare the means of the two sets of hardness data before and after being subjected to high temperature (HT). We assume that hardness follows a normal distribution, and we will use the difference in the sample means to calculate the test statistic, which follows a t-distribution under the null hypothesis.

Without the raw data calculations here, the test statistic's distribution would be t-distribution with degrees of freedom (df) equal to the number of paired samples minus one. To determine the exact p-value, one would typically perform the t-test calculations, which involve finding the mean difference, the standard deviation of the differences, and the standard error of the mean difference.

Then, the test statistic is compared against a t-distribution with the appropriate df to find the p-value.