The UCLA Internet Report (February 2003) estimated that a proportion of roughly 0.75 of online homes are still using dial-up access, but claimed that the use of dial-up is declining. Is that really the case? To examine this, a follow-up study was conducted a year later in which, out of a random sample of 1,308 households that had Internet access, 804 were connecting using a dial-up modem.Let p be the proportion of all U.S. Internet-using households who have dial-up access. In the previous activity, we established that the appropriate hypotheses here are:H0: p = 0.75 and Ha: p < 0.75
Question
The UCLA Internet Report (February 2003) estimated that a proportion of roughly 0.75 of online homes are still using dial-up access, but claimed that the use of dial-up is declining. Is that really the case? To examine this, a follow-up study was conducted a year later in which, out of a random sample of 1,308 households that had Internet access, 804 were connecting using a dial-up modem.Let p be the proportion of all U.S. Internet-using households who have dial-up access. In the previous activity, we established that the appropriate hypotheses here are:H0: p = 0.75 and Ha: p < 0.75
Solution
To determine if the use of dial-up is indeed declining, we can perform a hypothesis test using the data provided.
Step 1: State the hypotheses. The null hypothesis is that the proportion of households using dial-up access is equal to 0.75 (H0: p = 0.75). The alternative hypothesis is that the proportion is less than 0.75 (Ha: p < 0.75).
Step 2: Collect and summarize the data. In the follow-up study, out of 1,308 households, 804 were using dial-up access. So, the sample proportion (p̂) is 804/1308 = 0.6147.
Step 3: Use the sample data to calculate the test statistic. In this case, we would use a z-test for a proportion. The formula is (p̂ - p0) / sqrt((p0*(1 - p0))/n), where p0 is the proportion under the null hypothesis, which is 0.75, and n is the sample size, which is 1308.
Step 4: Determine the p-value. The p-value is the probability of observing a test statistic as extreme as the one calculated (or more extreme) under the null hypothesis. You would use a standard normal (Z) distribution table or a technology tool to find this.
Step 5: Make a decision. If the p-value is less than the significance level (usually 0.05), we reject the null hypothesis in favor of the alternative. This would suggest that the proportion of households using dial-up access is less than 0.75, indicating a decline in dial-up usage. If the p-value is greater than the significance level, we do not reject the null hypothesis. This would suggest that the proportion of households using dial-up access is not significantly less than 0.75, indicating no significant decline in dial-up usage.
Remember, this test does not prove causation, it only shows an association. Other factors could also be influencing the decline in dial-up usage.
Similar Questions
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