Student's T-Distribution and its Applications

The samples size of two types of electric bulbs were tested for life (in hours) and the following data were obtained.

Test the hypothesis that the population means are equal at $5\%$ level of significance and state the null hypothesis and alternative hypothesis.

The marks secured by $n_1=9$ students in Statistics and that of $n_2=12$ Students in Business Mathematics are given below:

Test whether the mean marks obtained by the students in Statistics and Business mathematics differ significantly at $1\%$ level of significance.

An IQ test was conducted to $5$ Persons before and after they were trained. The results are given below:

Test whether any change in IQ at $1\%$ level of significance.

The number of pages typed by $5$ DTP-operators for 1 hour in the morning sessions are $10,12, 13, 8, 9$, and the number of pages typed by them in the afternoon are $11, 15, 12, 10, 8$. Is there any significant difference in the mean number of pages typed?

Samples of two types of electric bulbs were tested for life (in hours) and the following data were obtained.

Test the hypothesis that the population means are equal at $5\%$ level of significance.

The heights (in feet) of $6$ rain trees in a town $A$ are $30,28, 29, 32, 31, 36$ and that of $8$ rain trees in another town $B$ are $35, 36, 37, 30, 32, 29, 35, 30$. Is there any significant difference in mean heights of rain trees?

The average run of cricket player from the past records is $80$. The recent scores of the player in $6$ test matches are $84$, $82$, $83$, $79$, $83$ and $85$. Test whether the average run is more than $80$

A random sample of $10$ packets containing cashew nuts weigh (in grams) $70$,$120$,$110$,$101$,$88$,$83$,$95$,$98$,$107$,$100$ each. Test whether the population mean weight of $100$ grams?

A random sample of ten students is taken and their marks in a particular subject are recorded. The average mark is $60$ with standard deviation $6.5$. Test the hypothesis that the average mark of students is $55$.

Based on small samples explain the procedure to test significance for equality of means of two normal populations.

Explain the testing procedure to test the normal population mean, when population variance is unknown

Write all the applications of $t$-distribution

List out the properties of $t$-distribution

Write the standard error of the difference between sample means.

To test the difference between normal population means, write the test statistic

When can a paired t-test be applied?

Write about the paired t-statistic.

What is student’s$t$-statistic

Which test will be more appropriate to test whether there is an increase in mean sales when a random sample of $6$ salesmen was selected and the value of their sales made before and after the training is recorded.

Find the standard deviation of the combined sample where ${s_1}^2$ and ${s_2}^2$ are respectively the variance of two independent random samples of sizes ‘$m$’ and ‘$n$'.