Question

A robotics company launches two types of home robots with different sizes. Management thinks that smaller models have more maintenance issues than the larger models. A marketing survey reveals that in sample of 45 robots with no maintenance issues the mean height was 61.4 inches. A second sample of 39 robots with maintenance issues had mean height of 60.6 inches. Assume that the population of heights with no issues has a population standard deviation of 1.2 inches while the population of those with issues has a standard deviation of 1.1 inches. We want to test the hypothesis that those with issues are smaller models . the significance level is 0.05. Based on this story below the critical value, the test statistic value and the decision are respectively:

Answer 1

Solution :

Null and alternative hypotheses :

The null and alternative hypotheses would be as follows :

Where, μ_{​​​​​​1} and μ_{​​​​​​2} are population mean sizes of robots with no maintenance issues and with maintenance issues respectively.

Test statistic :

To test the hypothesis we shall use two samples z-test for testing the equality of means. The test statistic is given as follows :

Where, x̅_{​​​​​1}, x̅_{​​​​​2} are sample means, σ_{​​​​​​1}, σ_{​​​​​​2} are population standard deviations and n_{​​​​​​1}, n_{​​​​​​2} are sample sizes.

We have following informations :

The value of the test statistic is 3.1866.

Critical value :

Significance level = 0.05

Our test is right-tailed test. So we shall obtain right-tailed critical value of Z at 0.05 significance level. The right-tailed critical z value at 0.05 significance level is given by,

Critical value = Z_(0.05) = 1.645

Decision :

(3.1866 > 1.645)

Since, value of the test statistic is greater than the critical value, therefore we shall reject the null hypothesis (H_{​​​​​​0}) at 0.05 significance level.

Conclusion : At 0.05 significance level, there is sufficient evidence to conclude that smaller models have more maintenance issues than the larger models.

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